mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -765,7 +765,7 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
   apply Nat.Coprime.symm
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
-  replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h
+  replace h := (ZMod.natCast_zmod_eq_zero_iff_dvd a p).2 h
   rw [is_root.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
   by_cases hone : n = 1
   · simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -583,7 +583,7 @@ theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [Com
           algebraMap _ _ (X ^ n - 1) :=
     by
     intro n hn
-    rw [← prod_cyclotomic_eq_X_pow_sub_one hn R, RingHom.map_prod]
+    rw [← prod_cyclotomic_eq_X_pow_sub_one hn R, map_prod]
   rw [(prod_eq_iff_prod_pow_moebius_eq_of_nonzero (fun n hn => _) fun n hn => _).1 h n hpos] <;>
     rw [Ne.def, IsFractionRing.to_map_eq_zero_iff]
   · apply cyclotomic_ne_zero

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -6,8 +6,8 @@ Authors: Riccardo Brasca
 import Algebra.NeZero
 import Algebra.Polynomial.BigOperators
 import RingTheory.RootsOfUnity.Complex
-import Data.Polynomial.Lifts
-import Data.Polynomial.Splits
+import Algebra.Polynomial.Lifts
+import Algebra.Polynomial.Splits
 import Data.ZMod.Algebra
 import FieldTheory.RatFunc
 import FieldTheory.Separable

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -8,8 +8,8 @@ import Algebra.Polynomial.BigOperators
 import RingTheory.RootsOfUnity.Complex
 import Data.Polynomial.Lifts
 import Data.Polynomial.Splits
-import Data.Zmod.Algebra
-import FieldTheory.Ratfunc
+import Data.ZMod.Algebra
+import FieldTheory.RatFunc
 import FieldTheory.Separable
 import NumberTheory.ArithmeticFunction
 import RingTheory.RootsOfUnity.Basic
@@ -711,7 +711,7 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
 theorem cyclotomic_prime_pow_mul_X_pow_sub_one (R : Type _) [CommRing R] (p k : ℕ)
     [hn : Fact (Nat.Prime p)] :
     cyclotomic (p ^ (k + 1)) R * (X ^ p ^ k - 1) = X ^ p ^ (k + 1) - 1 := by
-  rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, ← pow_mul, pow_succ, mul_comm]
+  rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, ← pow_mul, pow_succ', mul_comm]
 #align polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -256,7 +256,7 @@ theorem int_coeff_of_cyclotomic' {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
     intro x hx
     have xsmall := (Nat.mem_properDivisors.1 hx).2
     obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hx).1
-    rw [mul_comm] at hd 
+    rw [mul_comm] at hd
     exact ihk x xsmall (h.pow hpos hd)
   replace Bint := lifts_and_degree_eq_and_monic Bint Bmo
   obtain ⟨B₁, hB₁, hB₁deg, hB₁mo⟩ := Bint
@@ -652,7 +652,7 @@ theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type _} [CommRing K] [IsDom
     by
     intro i hi
     obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hi).1
-    rw [mul_comm] at hd 
+    rw [mul_comm] at hd
     exact hk i (Nat.mem_properDivisors.1 hi).2 (IsPrimitiveRoot.pow hpos hz hd)
   rw [@cyclotomic_eq_X_pow_sub_one_div _ _ _ hpos, cyclotomic'_eq_X_pow_sub_one_div hpos hz,
     Finset.prod_congr (refl k.proper_divisors) h]
@@ -696,13 +696,13 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
     by
     intro m
     have := eq_cyclotomic_iff (pow_pos hp.pos (m + 1)) _
-    rw [eq_comm] at this 
+    rw [eq_comm] at this
     rw [this, Nat.prod_properDivisors_prime_pow hp]
   induction' n with n_n n_ih
   · haveI := Fact.mk hp; simp [cyclotomic_prime]
   rw [((eq_cyclotomic_iff (pow_pos hp.pos (n_n.succ + 1)) _).mpr _).symm]
   rw [Nat.prod_properDivisors_prime_pow hp, Finset.prod_range_succ, n_ih]
-  rw [this] at n_ih 
+  rw [this] at n_ih
   rw [mul_comm _ (∑ i in _, _), n_ih, geom_sum_mul, sub_left_inj, ← pow_mul, pow_add, pow_one]
 #align polynomial.cyclotomic_prime_pow_eq_geom_sum Polynomial.cyclotomic_prime_pow_eq_geom_sum
 -/
@@ -751,7 +751,7 @@ theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
     rw [coeff_zero_eq_eval_zero _]
     simp only [zero_pow (lt_of_lt_of_le zero_lt_two hn), eval_X, eval_one, zero_sub, eval_pow,
       eval_sub]
-  rw [hzero] at heq 
+  rw [hzero] at heq
   exact neg_inj.mp (Eq.symm HEq)
 #align polynomial.cyclotomic_coeff_zero Polynomial.cyclotomic_coeff_zero
 -/
@@ -766,14 +766,14 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h
-  rw [is_root.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot 
+  rw [is_root.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
   by_cases hone : n = 1
   · simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
-      one_ne_zero, coeff_sub] at hroot 
+      one_ne_zero, coeff_sub] at hroot
     exact hroot
   rw [cyclotomic_coeff_zero (ZMod p)
       (Nat.succ_le_of_lt (lt_of_le_of_ne (Nat.succ_le_of_lt hpos) (Ne.symm hone)))] at
-    hroot 
+    hroot
   exact one_ne_zero hroot
 #align polynomial.coprime_of_root_cyclotomic Polynomial.coprime_of_root_cyclotomic
 -/
@@ -791,10 +791,10 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
   by
   apply orderOf_dvd_of_pow_eq_one
   suffices hpow : eval (Nat.castRingHom (ZMod p) a) (X ^ n - 1 : (ZMod p)[X]) = 0
-  · simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow 
+  · simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
     apply Units.val_eq_one.1
     simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
-  rw [is_root.def] at hroot 
+  rw [is_root.def] at hroot
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
     Finset.prod_cons, eval_mul, hroot, MulZeroClass.zero_mul]
 #align polynomial.order_of_root_cyclotomic_dvd Polynomial.orderOf_root_cyclotomic_dvd

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -562,7 +562,7 @@ theorem X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd (R) [CommRing R] {
 
 section ArithmeticFunction
 
-open Nat.ArithmeticFunction
+open ArithmeticFunction
 
 open scoped ArithmeticFunction
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -160,14 +160,14 @@ varies over the `n`-th roots of unity. -/
 theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     X ^ n - 1 = ∏ ζ in nthRootsFinset n R, (X - C ζ) :=
   by
-  rw [nth_roots_finset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]
+  rw [nth_roots_finset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_one_nodup h)]
   simp only [Finset.prod_mk, RingHom.map_one]
   rw [nth_roots]
   have hmonic : (X ^ n - C (1 : R)).Monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm
   symm
   apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmonic
   rw [@nat_degree_X_pow_sub_C R _ _ n 1, ← nth_roots]
-  exact IsPrimitiveRoot.card_nthRoots h
+  exact IsPrimitiveRoot.card_nthRoots_one h
 #align polynomial.X_pow_sub_one_eq_prod Polynomial.X_pow_sub_one_eq_prod
 -/
 
@@ -191,7 +191,8 @@ theorem cyclotomic'_splits (n : ℕ) : Splits (RingHom.id K) (cyclotomic' n K) :
 /-- If there is a primitive `n`-th root of unity in `K`, then `X ^ n - 1`splits. -/
 theorem X_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
     Splits (RingHom.id K) (X ^ n - C (1 : K)) := by
-  rw [splits_iff_card_roots, ← nth_roots, IsPrimitiveRoot.card_nthRoots h, nat_degree_X_pow_sub_C]
+  rw [splits_iff_card_roots, ← nth_roots, IsPrimitiveRoot.card_nthRoots_one h,
+    nat_degree_X_pow_sub_C]
 #align polynomial.X_pow_sub_one_splits Polynomial.X_pow_sub_one_splits
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,16 +3,16 @@ Copyright (c) 2020 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 -/
-import Mathbin.Algebra.NeZero
-import Mathbin.Algebra.Polynomial.BigOperators
-import Mathbin.RingTheory.RootsOfUnity.Complex
-import Mathbin.Data.Polynomial.Lifts
-import Mathbin.Data.Polynomial.Splits
-import Mathbin.Data.Zmod.Algebra
-import Mathbin.FieldTheory.Ratfunc
-import Mathbin.FieldTheory.Separable
-import Mathbin.NumberTheory.ArithmeticFunction
-import Mathbin.RingTheory.RootsOfUnity.Basic
+import Algebra.NeZero
+import Algebra.Polynomial.BigOperators
+import RingTheory.RootsOfUnity.Complex
+import Data.Polynomial.Lifts
+import Data.Polynomial.Splits
+import Data.Zmod.Algebra
+import FieldTheory.Ratfunc
+import FieldTheory.Separable
+import NumberTheory.ArithmeticFunction
+import RingTheory.RootsOfUnity.Basic
 
 #align_import ring_theory.polynomial.cyclotomic.basic from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -759,9 +759,9 @@ theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
 /-- If `(a : ℕ)` is a root of `cyclotomic n (zmod p)`, where `p` is a prime, then `a` and `p` are
 coprime. -/
 theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime : Fact p.Prime] {a : ℕ}
-    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.coprime p :=
+    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.Coprime p :=
   by
-  apply Nat.coprime.symm
+  apply Nat.Coprime.symm
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -548,7 +548,7 @@ theorem X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd (R) [CommRing R] {
     (h : d ∈ n.properDivisors) : (X ^ d - 1) * cyclotomic n R ∣ X ^ n - 1 :=
   by
   have hdn := (nat.mem_proper_divisors.mp h).2
-  use ∏ x in n.proper_divisors \ d.divisors, cyclotomic x R
+  use∏ x in n.proper_divisors \ d.divisors, cyclotomic x R
   symm
   convert X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd R h using 1
   rw [mul_assoc]

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.cyclotomic.basic
-! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.NeZero
 import Mathbin.Algebra.Polynomial.BigOperators
@@ -19,6 +14,8 @@ import Mathbin.FieldTheory.Separable
 import Mathbin.NumberTheory.ArithmeticFunction
 import Mathbin.RingTheory.RootsOfUnity.Basic
 
+#align_import ring_theory.polynomial.cyclotomic.basic from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
+
 /-!
 # Cyclotomic polynomials.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -97,6 +97,7 @@ theorem cyclotomic'_one (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 1 R
 #align polynomial.cyclotomic'_one Polynomial.cyclotomic'_one
 -/
 
+#print Polynomial.cyclotomic'_two /-
 /-- The second modified cyclotomic polyomial is `X + 1` if the characteristic of `R` is not `2`. -/
 @[simp]
 theorem cyclotomic'_two (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [CharP R p] (hp : p ≠ 2) :
@@ -108,6 +109,7 @@ theorem cyclotomic'_two (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [CharP
     exact ⟨IsPrimitiveRoot.neg_one p hp, fun x => IsPrimitiveRoot.eq_neg_one_of_two_right⟩
   simp only [prim_root_two, Finset.prod_singleton, RingHom.map_neg, RingHom.map_one, sub_neg_eq_add]
 #align polynomial.cyclotomic'_two Polynomial.cyclotomic'_two
+-/
 
 #print Polynomial.cyclotomic'.monic /-
 /-- `cyclotomic' n R` is monic. -/
@@ -155,6 +157,7 @@ theorem roots_of_cyclotomic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
 #align polynomial.roots_of_cyclotomic Polynomial.roots_of_cyclotomic
 -/
 
+#print Polynomial.X_pow_sub_one_eq_prod /-
 /-- If there is a primitive `n`th root of unity in `K`, then `X ^ n - 1 = ∏ (X - μ)`, where `μ`
 varies over the `n`-th roots of unity. -/
 theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
@@ -169,6 +172,7 @@ theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitive
   rw [@nat_degree_X_pow_sub_C R _ _ n 1, ← nth_roots]
   exact IsPrimitiveRoot.card_nthRoots h
 #align polynomial.X_pow_sub_one_eq_prod Polynomial.X_pow_sub_one_eq_prod
+-/
 
 end IsDomain
 
@@ -186,12 +190,15 @@ theorem cyclotomic'_splits (n : ℕ) : Splits (RingHom.id K) (cyclotomic' n K) :
 #align polynomial.cyclotomic'_splits Polynomial.cyclotomic'_splits
 -/
 
+#print Polynomial.X_pow_sub_one_splits /-
 /-- If there is a primitive `n`-th root of unity in `K`, then `X ^ n - 1`splits. -/
 theorem X_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
     Splits (RingHom.id K) (X ^ n - C (1 : K)) := by
   rw [splits_iff_card_roots, ← nth_roots, IsPrimitiveRoot.card_nthRoots h, nat_degree_X_pow_sub_C]
 #align polynomial.X_pow_sub_one_splits Polynomial.X_pow_sub_one_splits
+-/
 
+#print Polynomial.prod_cyclotomic'_eq_X_pow_sub_one /-
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `∏ i in nat.divisors n, cyclotomic' i K = X ^ n - 1`. -/
 theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
@@ -202,7 +209,9 @@ theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type _} [CommRing K] [IsDomain K]
   simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← Finset.prod_biUnion hd,
     h.nth_roots_one_eq_bUnion_primitive_roots]
 #align polynomial.prod_cyclotomic'_eq_X_pow_sub_one Polynomial.prod_cyclotomic'_eq_X_pow_sub_one
+-/
 
+#print Polynomial.cyclotomic'_eq_X_pow_sub_one_div /-
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `cyclotomic' n K = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic' i K)`. -/
 theorem cyclotomic'_eq_X_pow_sub_one_div {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
@@ -223,7 +232,9 @@ theorem cyclotomic'_eq_X_pow_sub_one_div {K : Type _} [CommRing K] [IsDomain K]
   intro h
   exact monic.ne_zero prod_monic (degree_eq_bot.1 h)
 #align polynomial.cyclotomic'_eq_X_pow_sub_one_div Polynomial.cyclotomic'_eq_X_pow_sub_one_div
+-/
 
+#print Polynomial.int_coeff_of_cyclotomic' /-
 /-- If there is a primitive `n`-th root of unity in `K`, then `cyclotomic' n K` comes from a
 monic polynomial with integer coefficients. -/
 theorem int_coeff_of_cyclotomic' {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
@@ -265,7 +276,9 @@ theorem int_coeff_of_cyclotomic' {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
   rw [coe_map_ring_hom, map_div_by_monic (Int.castRingHom K) hB₁mo, hB₁, ← huniq.1]
   simp
 #align polynomial.int_coeff_of_cyclotomic' Polynomial.int_coeff_of_cyclotomic'
+-/
 
+#print Polynomial.unique_int_coeff_of_cycl /-
 /-- If `K` is of characteristic `0` and there is a primitive `n`-th root of unity in `K`,
 then `cyclotomic n K` comes from a unique polynomial with integer coefficients. -/
 theorem unique_int_coeff_of_cycl {K : Type _} [CommRing K] [IsDomain K] [CharZero K] {ζ : K}
@@ -276,6 +289,7 @@ theorem unique_int_coeff_of_cycl {K : Type _} [CommRing K] [IsDomain K] [CharZer
   apply map_injective (Int.castRingHom K) Int.cast_injective
   rw [hP.1, hQ]
 #align polynomial.unique_int_coeff_of_cycl Polynomial.unique_int_coeff_of_cycl
+-/
 
 end Field
 
@@ -291,6 +305,7 @@ def cyclotomic (n : ℕ) (R : Type _) [Ring R] : R[X] :=
 #align polynomial.cyclotomic Polynomial.cyclotomic
 -/
 
+#print Polynomial.int_cyclotomic_rw /-
 theorem int_cyclotomic_rw {n : ℕ} (h : n ≠ 0) :
     cyclotomic n ℤ = (int_coeff_of_cyclotomic' (Complex.isPrimitiveRoot_exp n h)).some :=
   by
@@ -298,7 +313,9 @@ theorem int_cyclotomic_rw {n : ℕ} (h : n ≠ 0) :
   ext i
   simp only [coeff_map, Int.cast_id, eq_intCast]
 #align polynomial.int_cyclotomic_rw Polynomial.int_cyclotomic_rw
+-/
 
+#print Polynomial.map_cyclotomic_int /-
 /-- `cyclotomic n R` comes from `cyclotomic n ℤ`. -/
 theorem map_cyclotomic_int (n : ℕ) (R : Type _) [Ring R] :
     map (Int.castRingHom R) (cyclotomic n ℤ) = cyclotomic n R :=
@@ -307,7 +324,9 @@ theorem map_cyclotomic_int (n : ℕ) (R : Type _) [Ring R] :
   · simp only [hzero, cyclotomic, dif_pos, Polynomial.map_one]
   simp only [cyclotomic, int_cyclotomic_rw, hzero, Ne.def, dif_neg, not_false_iff]
 #align polynomial.map_cyclotomic_int Polynomial.map_cyclotomic_int
+-/
 
+#print Polynomial.int_cyclotomic_spec /-
 theorem int_cyclotomic_spec (n : ℕ) :
     map (Int.castRingHom ℂ) (cyclotomic n ℤ) = cyclotomic' n ℂ ∧
       (cyclotomic n ℤ).degree = (cyclotomic' n ℂ).degree ∧ (cyclotomic n ℤ).Monic :=
@@ -319,14 +338,18 @@ theorem int_cyclotomic_spec (n : ℕ) :
   rw [int_cyclotomic_rw hzero]
   exact (int_coeff_of_cyclotomic' (Complex.isPrimitiveRoot_exp n hzero)).choose_spec
 #align polynomial.int_cyclotomic_spec Polynomial.int_cyclotomic_spec
+-/
 
+#print Polynomial.int_cyclotomic_unique /-
 theorem int_cyclotomic_unique {n : ℕ} {P : ℤ[X]} (h : map (Int.castRingHom ℂ) P = cyclotomic' n ℂ) :
     P = cyclotomic n ℤ :=
   by
   apply map_injective (Int.castRingHom ℂ) Int.cast_injective
   rw [h, (int_cyclotomic_spec n).1]
 #align polynomial.int_cyclotomic_unique Polynomial.int_cyclotomic_unique
+-/
 
+#print Polynomial.map_cyclotomic /-
 /-- The definition of `cyclotomic n R` commutes with any ring homomorphism. -/
 @[simp]
 theorem map_cyclotomic (n : ℕ) {R S : Type _} [Ring R] [Ring S] (f : R →+* S) :
@@ -335,11 +358,14 @@ theorem map_cyclotomic (n : ℕ) {R S : Type _} [Ring R] [Ring S] (f : R →+* S
   rw [← map_cyclotomic_int n R, ← map_cyclotomic_int n S, map_map]
   congr
 #align polynomial.map_cyclotomic Polynomial.map_cyclotomic
+-/
 
+#print Polynomial.cyclotomic.eval_apply /-
 theorem cyclotomic.eval_apply {R S : Type _} (q : R) (n : ℕ) [Ring R] [Ring S] (f : R →+* S) :
     eval (f q) (cyclotomic n S) = f (eval q (cyclotomic n R)) := by
   rw [← map_cyclotomic n f, eval_map, eval₂_at_apply]
 #align polynomial.cyclotomic.eval_apply Polynomial.cyclotomic.eval_apply
+-/
 
 #print Polynomial.cyclotomic_zero /-
 /-- The zeroth cyclotomic polyomial is `1`. -/
@@ -419,6 +445,7 @@ theorem degree_cyclotomic_pos (n : ℕ) (R : Type _) (hpos : 0 < n) [Ring R] [No
 
 open Finset
 
+#print Polynomial.prod_cyclotomic_eq_X_pow_sub_one /-
 /-- `∏ i in nat.divisors n, cyclotomic i R = X ^ n - 1`. -/
 theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [CommRing R] :
     ∏ i in Nat.divisors n, cyclotomic i R = X ^ n - 1 :=
@@ -432,7 +459,9 @@ theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [
   simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one,
     Polynomial.map_pow, Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) integer
 #align polynomial.prod_cyclotomic_eq_X_pow_sub_one Polynomial.prod_cyclotomic_eq_X_pow_sub_one
+-/
 
+#print Polynomial.cyclotomic.dvd_X_pow_sub_one /-
 theorem cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomic n R ∣ X ^ n - 1 :=
   by
   suffices cyclotomic n ℤ ∣ X ^ n - 1 by
@@ -443,7 +472,9 @@ theorem cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomi
   rw [← prod_cyclotomic_eq_X_pow_sub_one hn]
   exact Finset.dvd_prod_of_mem _ (n.mem_divisors_self hn.ne')
 #align polynomial.cyclotomic.dvd_X_pow_sub_one Polynomial.cyclotomic.dvd_X_pow_sub_one
+-/
 
+#print Polynomial.prod_cyclotomic_eq_geom_sum /-
 theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
     ∏ i in n.divisors.eraseₓ 1, cyclotomic i R = ∑ i in Finset.range n, X ^ i :=
   by
@@ -453,7 +484,9 @@ theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
   rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (Nat.one_mem_divisors.2 h.ne'),
     cyclotomic_one, geom_sum_mul, prod_cyclotomic_eq_X_pow_sub_one h]
 #align polynomial.prod_cyclotomic_eq_geom_sum Polynomial.prod_cyclotomic_eq_geom_sum
+-/
 
+#print Polynomial.cyclotomic_prime /-
 /-- If `p` is prime, then `cyclotomic p R = ∑ i in range p, X ^ i`. -/
 theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
     cyclotomic p R = ∑ i in Finset.range p, X ^ i :=
@@ -464,10 +497,13 @@ theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
   rw [← prod_cyclotomic_eq_geom_sum hp.out.pos, hp.out.divisors,
     erase_insert (mem_singleton.not.2 hp.out.ne_one.symm), prod_singleton]
 #align polynomial.cyclotomic_prime Polynomial.cyclotomic_prime
+-/
 
+#print Polynomial.cyclotomic_prime_mul_X_sub_one /-
 theorem cyclotomic_prime_mul_X_sub_one (R : Type _) [Ring R] (p : ℕ) [hn : Fact (Nat.Prime p)] :
     cyclotomic p R * (X - 1) = X ^ p - 1 := by rw [cyclotomic_prime, geom_sum_mul]
 #align polynomial.cyclotomic_prime_mul_X_sub_one Polynomial.cyclotomic_prime_mul_X_sub_one
+-/
 
 #print Polynomial.cyclotomic_two /-
 @[simp]
@@ -475,11 +511,14 @@ theorem cyclotomic_two (R : Type _) [Ring R] : cyclotomic 2 R = X + 1 := by simp
 #align polynomial.cyclotomic_two Polynomial.cyclotomic_two
 -/
 
+#print Polynomial.cyclotomic_three /-
 @[simp]
 theorem cyclotomic_three (R : Type _) [Ring R] : cyclotomic 3 R = X ^ 2 + X + 1 := by
   simp [cyclotomic_prime, sum_range_succ']
 #align polynomial.cyclotomic_three Polynomial.cyclotomic_three
+-/
 
+#print Polynomial.cyclotomic_dvd_geom_sum_of_dvd /-
 theorem cyclotomic_dvd_geom_sum_of_dvd (R) [Ring R] {d n : ℕ} (hdn : d ∣ n) (hd : d ≠ 1) :
     cyclotomic d R ∣ ∑ i in Finset.range n, X ^ i :=
   by
@@ -492,7 +531,9 @@ theorem cyclotomic_dvd_geom_sum_of_dvd (R) [Ring R] {d n : ℕ} (hdn : d ∣ n)
   apply Finset.dvd_prod_of_mem
   simp [hd, hdn, hn.ne']
 #align polynomial.cyclotomic_dvd_geom_sum_of_dvd Polynomial.cyclotomic_dvd_geom_sum_of_dvd
+-/
 
+#print Polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd /-
 theorem X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
     (h : d ∈ n.properDivisors) :
     (X ^ d - 1) * ∏ x in n.divisors \ d.divisors, cyclotomic x R = X ^ n - 1 :=
@@ -503,7 +544,9 @@ theorem X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd (R) [CommRing
   rw [← prod_cyclotomic_eq_X_pow_sub_one h0d, ← prod_cyclotomic_eq_X_pow_sub_one h0n, mul_comm,
     Finset.prod_sdiff (Nat.divisors_subset_of_dvd h0n.ne' hd)]
 #align polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd Polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd
+-/
 
+#print Polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd /-
 theorem X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
     (h : d ∈ n.properDivisors) : (X ^ d - 1) * cyclotomic n R ∣ X ^ n - 1 :=
   by
@@ -517,6 +560,7 @@ theorem X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd (R) [CommRing R] {
   · exact Finset.not_mem_sdiff_of_not_mem_left Nat.properDivisors.not_self_mem
   · exact fun hk => hdn.not_le <| Nat.divisor_le hk
 #align polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd Polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd
+-/
 
 section ArithmeticFunction
 
@@ -524,6 +568,7 @@ open Nat.ArithmeticFunction
 
 open scoped ArithmeticFunction
 
+#print Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius /-
 /-- `cyclotomic n R` can be expressed as a product in a fraction field of `R[X]`
   using Möbius inversion. -/
 theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [CommRing R]
@@ -547,9 +592,11 @@ theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [Com
   · apply monic.ne_zero
     apply monic_X_pow_sub_C _ (ne_of_gt hn)
 #align polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius
+-/
 
 end ArithmeticFunction
 
+#print Polynomial.cyclotomic_eq_X_pow_sub_one_div /-
 /-- We have
 `cyclotomic n R = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic i K)`. -/
 theorem cyclotomic_eq_X_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) :
@@ -571,7 +618,9 @@ theorem cyclotomic_eq_X_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpo
   intro h
   exact monic.ne_zero prod_monic (degree_eq_bot.1 h)
 #align polynomial.cyclotomic_eq_X_pow_sub_one_div Polynomial.cyclotomic_eq_X_pow_sub_one_div
+-/
 
+#print Polynomial.X_pow_sub_one_dvd_prod_cyclotomic /-
 /-- If `m` is a proper divisor of `n`, then `X ^ m - 1` divides
 `∏ i in nat.proper_divisors n, cyclotomic i R`. -/
 theorem X_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ} (hpos : 0 < n)
@@ -588,7 +637,9 @@ theorem X_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ}
     prod_cyclotomic_eq_X_pow_sub_one (Nat.pos_of_mem_properDivisors hm)]
   exact ⟨∏ x : ℕ in n.proper_divisors \ m.divisors, cyclotomic x R, by rw [mul_comm]⟩
 #align polynomial.X_pow_sub_one_dvd_prod_cyclotomic Polynomial.X_pow_sub_one_dvd_prod_cyclotomic
+-/
 
+#print Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots /-
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `cyclotomic n K = ∏ μ in primitive_roots n R, (X - C μ)`. In particular,
 `cyclotomic n K = cyclotomic' n K` -/
@@ -608,7 +659,9 @@ theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type _} [CommRing K] [IsDom
   rw [@cyclotomic_eq_X_pow_sub_one_div _ _ _ hpos, cyclotomic'_eq_X_pow_sub_one_div hpos hz,
     Finset.prod_congr (refl k.proper_divisors) h]
 #align polynomial.cyclotomic_eq_prod_X_sub_primitive_roots Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots
+-/
 
+#print Polynomial.eq_cyclotomic_iff /-
 theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P : R[X]) :
     P = cyclotomic n R ↔ P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R = X ^ n - 1 :=
   by
@@ -628,7 +681,9 @@ theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P
     intro h
     exact monic.ne_zero prod_monic (degree_eq_bot.1 h)
 #align polynomial.eq_cyclotomic_iff Polynomial.eq_cyclotomic_iff
+-/
 
+#print Polynomial.cyclotomic_prime_pow_eq_geom_sum /-
 /-- If `p ^ k` is a prime power, then
 `cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ (p ^ n)) ^ i`. -/
 theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (hp : p.Prime) :
@@ -652,13 +707,17 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
   rw [this] at n_ih 
   rw [mul_comm _ (∑ i in _, _), n_ih, geom_sum_mul, sub_left_inj, ← pow_mul, pow_add, pow_one]
 #align polynomial.cyclotomic_prime_pow_eq_geom_sum Polynomial.cyclotomic_prime_pow_eq_geom_sum
+-/
 
+#print Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one /-
 theorem cyclotomic_prime_pow_mul_X_pow_sub_one (R : Type _) [CommRing R] (p k : ℕ)
     [hn : Fact (Nat.Prime p)] :
     cyclotomic (p ^ (k + 1)) R * (X ^ p ^ k - 1) = X ^ p ^ (k + 1) - 1 := by
   rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, ← pow_mul, pow_succ, mul_comm]
 #align polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one
+-/
 
+#print Polynomial.cyclotomic_coeff_zero /-
 /-- The constant term of `cyclotomic n R` is `1` if `2 ≤ n`. -/
 theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
     (cyclotomic n R).coeff 0 = 1 :=
@@ -697,7 +756,9 @@ theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
   rw [hzero] at heq 
   exact neg_inj.mp (Eq.symm HEq)
 #align polynomial.cyclotomic_coeff_zero Polynomial.cyclotomic_coeff_zero
+-/
 
+#print Polynomial.coprime_of_root_cyclotomic /-
 /-- If `(a : ℕ)` is a root of `cyclotomic n (zmod p)`, where `p` is a prime, then `a` and `p` are
 coprime. -/
 theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime : Fact p.Prime] {a : ℕ}
@@ -717,11 +778,13 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
     hroot 
   exact one_ne_zero hroot
 #align polynomial.coprime_of_root_cyclotomic Polynomial.coprime_of_root_cyclotomic
+-/
 
 end Cyclotomic
 
 section Order
 
+#print Polynomial.orderOf_root_cyclotomic_dvd /-
 /-- If `(a : ℕ)` is a root of `cyclotomic n (zmod p)`, then the multiplicative order of `a` modulo
 `p` divides `n`. -/
 theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.Prime] {a : ℕ}
@@ -737,6 +800,7 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
     Finset.prod_cons, eval_mul, hroot, MulZeroClass.zero_mul]
 #align polynomial.order_of_root_cyclotomic_dvd Polynomial.orderOf_root_cyclotomic_dvd
+-/
 
 end Order
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -76,7 +76,7 @@ variable {R : Type _} [CommRing R] [IsDomain R]
 /-- The modified `n`-th cyclotomic polynomial with coefficients in `R`, it is the usual cyclotomic
 polynomial if there is a primitive `n`-th root of unity in `R`. -/
 def cyclotomic' (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] : R[X] :=
-  ∏ μ in primitiveRoots n R, X - C μ
+  ∏ μ in primitiveRoots n R, (X - C μ)
 #align polynomial.cyclotomic' Polynomial.cyclotomic'
 -/
 
@@ -158,7 +158,7 @@ theorem roots_of_cyclotomic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
 /-- If there is a primitive `n`th root of unity in `K`, then `X ^ n - 1 = ∏ (X - μ)`, where `μ`
 varies over the `n`-th roots of unity. -/
 theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
-    X ^ n - 1 = ∏ ζ in nthRootsFinset n R, X - C ζ :=
+    X ^ n - 1 = ∏ ζ in nthRootsFinset n R, (X - C ζ) :=
   by
   rw [nth_roots_finset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]
   simp only [Finset.prod_mk, RingHom.map_one]
@@ -195,8 +195,7 @@ theorem X_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `∏ i in nat.divisors n, cyclotomic' i K = X ^ n - 1`. -/
 theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
-    (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
-    (∏ i in Nat.divisors n, cyclotomic' i K) = X ^ n - 1 :=
+    (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) : ∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1 :=
   by
   have hd : (n.divisors : Set ℕ).PairwiseDisjoint fun k => primitiveRoots k K :=
     fun x hx y hy hne => IsPrimitiveRoot.disjoint hne
@@ -422,9 +421,9 @@ open Finset
 
 /-- `∏ i in nat.divisors n, cyclotomic i R = X ^ n - 1`. -/
 theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [CommRing R] :
-    (∏ i in Nat.divisors n, cyclotomic i R) = X ^ n - 1 :=
+    ∏ i in Nat.divisors n, cyclotomic i R = X ^ n - 1 :=
   by
-  have integer : (∏ i in Nat.divisors n, cyclotomic i ℤ) = X ^ n - 1 :=
+  have integer : ∏ i in Nat.divisors n, cyclotomic i ℤ = X ^ n - 1 :=
     by
     apply map_injective (Int.castRingHom ℂ) Int.cast_injective
     simp only [Polynomial.map_prod, int_cyclotomic_spec, Polynomial.map_pow, map_X,
@@ -446,9 +445,9 @@ theorem cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomi
 #align polynomial.cyclotomic.dvd_X_pow_sub_one Polynomial.cyclotomic.dvd_X_pow_sub_one
 
 theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
-    (∏ i in n.divisors.eraseₓ 1, cyclotomic i R) = ∑ i in Finset.range n, X ^ i :=
+    ∏ i in n.divisors.eraseₓ 1, cyclotomic i R = ∑ i in Finset.range n, X ^ i :=
   by
-  suffices (∏ i in n.divisors.eraseₓ 1, cyclotomic i ℤ) = ∑ i in Finset.range n, X ^ i by
+  suffices ∏ i in n.divisors.eraseₓ 1, cyclotomic i ℤ = ∑ i in Finset.range n, X ^ i by
     simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
       Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this
   rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (Nat.one_mem_divisors.2 h.ne'),
@@ -496,7 +495,7 @@ theorem cyclotomic_dvd_geom_sum_of_dvd (R) [Ring R] {d n : ℕ} (hdn : d ∣ n)
 
 theorem X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
     (h : d ∈ n.properDivisors) :
-    ((X ^ d - 1) * ∏ x in n.divisors \ d.divisors, cyclotomic x R) = X ^ n - 1 :=
+    (X ^ d - 1) * ∏ x in n.divisors \ d.divisors, cyclotomic x R = X ^ n - 1 :=
   by
   obtain ⟨hd, hdn⟩ := nat.mem_proper_divisors.mp h
   have h0n : 0 < n := pos_of_gt hdn
@@ -537,7 +536,7 @@ theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [Com
   have h :
     ∀ n : ℕ,
       0 < n →
-        (∏ i in Nat.divisors n, algebraMap _ (RatFunc R) (cyclotomic i R)) =
+        ∏ i in Nat.divisors n, algebraMap _ (RatFunc R) (cyclotomic i R) =
           algebraMap _ _ (X ^ n - 1) :=
     by
     intro n hn
@@ -594,7 +593,7 @@ theorem X_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ}
 `cyclotomic n K = ∏ μ in primitive_roots n R, (X - C μ)`. In particular,
 `cyclotomic n K = cyclotomic' n K` -/
 theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
-    {n : ℕ} (hz : IsPrimitiveRoot ζ n) : cyclotomic n K = ∏ μ in primitiveRoots n K, X - C μ :=
+    {n : ℕ} (hz : IsPrimitiveRoot ζ n) : cyclotomic n K = ∏ μ in primitiveRoots n K, (X - C μ) :=
   by
   rw [← cyclotomic']
   induction' n using Nat.strong_induction_on with k hk generalizing ζ hz
@@ -611,7 +610,7 @@ theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type _} [CommRing K] [IsDom
 #align polynomial.cyclotomic_eq_prod_X_sub_primitive_roots Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots
 
 theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P : R[X]) :
-    P = cyclotomic n R ↔ (P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R) = X ^ n - 1 :=
+    P = cyclotomic n R ↔ P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R = X ^ n - 1 :=
   by
   nontriviality R
   refine' ⟨fun hcycl => _, fun hP => _⟩
@@ -637,9 +636,9 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
   by
   have :
     ∀ m,
-      (cyclotomic (p ^ (m + 1)) R = ∑ i in Finset.range p, (X ^ p ^ m) ^ i) ↔
-        ((∑ i in Finset.range p, (X ^ p ^ m) ^ i) *
-            ∏ x : ℕ in Finset.range (m + 1), cyclotomic (p ^ x) R) =
+      cyclotomic (p ^ (m + 1)) R = ∑ i in Finset.range p, (X ^ p ^ m) ^ i ↔
+        (∑ i in Finset.range p, (X ^ p ^ m) ^ i) *
+            ∏ x : ℕ in Finset.range (m + 1), cyclotomic (p ^ x) R =
           X ^ p ^ (m + 1) - 1 :=
     by
     intro m
@@ -665,7 +664,7 @@ theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
     (cyclotomic n R).coeff 0 = 1 :=
   by
   induction' n using Nat.strong_induction_on with n hi
-  have hprod : (∏ i in Nat.properDivisors n, (Polynomial.cyclotomic i R).coeff 0) = -1 :=
+  have hprod : ∏ i in Nat.properDivisors n, (Polynomial.cyclotomic i R).coeff 0 = -1 :=
     by
     rw [←
       Finset.insert_erase (Nat.one_mem_properDivisors_iff_one_lt.2 (lt_of_lt_of_le one_lt_two hn)),
@@ -681,7 +680,7 @@ theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
       by
       intro j hj
       exact hi j (Nat.mem_properDivisors.1 (Finset.mem_erase.1 hj).2).2 (hleq j hj)
-    have hrw : (∏ x : ℕ in n.proper_divisors.erase 1, (cyclotomic x R).coeff 0) = 1 :=
+    have hrw : ∏ x : ℕ in n.proper_divisors.erase 1, (cyclotomic x R).coeff 0 = 1 :=
       by
       rw [Finset.prod_congr (refl (n.proper_divisors.erase 1)) hcongr]
       simp only [Finset.prod_const_one]

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.cyclotomic.basic
-! leanprover-community/mathlib commit 7fdeecc0d03cd40f7a165e6cf00a4d2286db599f
+! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -22,6 +22,9 @@ import Mathbin.RingTheory.RootsOfUnity.Basic
 /-!
 # Cyclotomic polynomials.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 For `n : ℕ` and an integral domain `R`, we define a modified version of the `n`-th cyclotomic
 polynomial with coefficients in `R`, denoted `cyclotomic' n R`, as `∏ (X - μ)`, where `μ` varies
 over the primitive `n`th roots of unity. If there is a primitive `n`th root of unity in `R` then

mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3

@@ -69,24 +69,30 @@ section IsDomain
 
 variable {R : Type _} [CommRing R] [IsDomain R]
 
+#print Polynomial.cyclotomic' /-
 /-- The modified `n`-th cyclotomic polynomial with coefficients in `R`, it is the usual cyclotomic
 polynomial if there is a primitive `n`-th root of unity in `R`. -/
 def cyclotomic' (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] : R[X] :=
   ∏ μ in primitiveRoots n R, X - C μ
 #align polynomial.cyclotomic' Polynomial.cyclotomic'
+-/
 
+#print Polynomial.cyclotomic'_zero /-
 /-- The zeroth modified cyclotomic polyomial is `1`. -/
 @[simp]
 theorem cyclotomic'_zero (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 0 R = 1 := by
   simp only [cyclotomic', Finset.prod_empty, primitiveRoots_zero]
 #align polynomial.cyclotomic'_zero Polynomial.cyclotomic'_zero
+-/
 
+#print Polynomial.cyclotomic'_one /-
 /-- The first modified cyclotomic polyomial is `X - 1`. -/
 @[simp]
 theorem cyclotomic'_one (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 1 R = X - 1 := by
   simp only [cyclotomic', Finset.prod_singleton, RingHom.map_one,
     IsPrimitiveRoot.primitiveRoots_one]
 #align polynomial.cyclotomic'_one Polynomial.cyclotomic'_one
+-/
 
 /-- The second modified cyclotomic polyomial is `X + 1` if the characteristic of `R` is not `2`. -/
 @[simp]
@@ -100,17 +106,22 @@ theorem cyclotomic'_two (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [CharP
   simp only [prim_root_two, Finset.prod_singleton, RingHom.map_neg, RingHom.map_one, sub_neg_eq_add]
 #align polynomial.cyclotomic'_two Polynomial.cyclotomic'_two
 
+#print Polynomial.cyclotomic'.monic /-
 /-- `cyclotomic' n R` is monic. -/
 theorem cyclotomic'.monic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
     (cyclotomic' n R).Monic :=
   monic_prod_of_monic _ _ fun z hz => monic_X_sub_C _
 #align polynomial.cyclotomic'.monic Polynomial.cyclotomic'.monic
+-/
 
+#print Polynomial.cyclotomic'_ne_zero /-
 /-- `cyclotomic' n R` is different from `0`. -/
 theorem cyclotomic'_ne_zero (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' n R ≠ 0 :=
   (cyclotomic'.monic n R).NeZero
 #align polynomial.cyclotomic'_ne_zero Polynomial.cyclotomic'_ne_zero
+-/
 
+#print Polynomial.natDegree_cyclotomic' /-
 /-- The natural degree of `cyclotomic' n R` is `totient n` if there is a primitive root of
 unity in `R`. -/
 theorem natDegree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
@@ -123,22 +134,27 @@ theorem natDegree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
   intro z hz
   exact X_sub_C_ne_zero z
 #align polynomial.nat_degree_cyclotomic' Polynomial.natDegree_cyclotomic'
+-/
 
+#print Polynomial.degree_cyclotomic' /-
 /-- The degree of `cyclotomic' n R` is `totient n` if there is a primitive root of unity in `R`. -/
 theorem degree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
     (cyclotomic' n R).degree = Nat.totient n := by
   simp only [degree_eq_nat_degree (cyclotomic'_ne_zero n R), nat_degree_cyclotomic' h]
 #align polynomial.degree_cyclotomic' Polynomial.degree_cyclotomic'
+-/
 
+#print Polynomial.roots_of_cyclotomic /-
 /-- The roots of `cyclotomic' n R` are the primitive `n`-th roots of unity. -/
 theorem roots_of_cyclotomic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
     (cyclotomic' n R).roots = (primitiveRoots n R).val := by rw [cyclotomic'];
   exact roots_prod_X_sub_C (primitiveRoots n R)
 #align polynomial.roots_of_cyclotomic Polynomial.roots_of_cyclotomic
+-/
 
 /-- If there is a primitive `n`th root of unity in `K`, then `X ^ n - 1 = ∏ (X - μ)`, where `μ`
 varies over the `n`-th roots of unity. -/
-theorem x_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
+theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     X ^ n - 1 = ∏ ζ in nthRootsFinset n R, X - C ζ :=
   by
   rw [nth_roots_finset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]
@@ -149,7 +165,7 @@ theorem x_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitive
   apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmonic
   rw [@nat_degree_X_pow_sub_C R _ _ n 1, ← nth_roots]
   exact IsPrimitiveRoot.card_nthRoots h
-#align polynomial.X_pow_sub_one_eq_prod Polynomial.x_pow_sub_one_eq_prod
+#align polynomial.X_pow_sub_one_eq_prod Polynomial.X_pow_sub_one_eq_prod
 
 end IsDomain
 
@@ -157,6 +173,7 @@ section Field
 
 variable {K : Type _} [Field K]
 
+#print Polynomial.cyclotomic'_splits /-
 /-- `cyclotomic' n K` splits. -/
 theorem cyclotomic'_splits (n : ℕ) : Splits (RingHom.id K) (cyclotomic' n K) :=
   by
@@ -164,16 +181,17 @@ theorem cyclotomic'_splits (n : ℕ) : Splits (RingHom.id K) (cyclotomic' n K) :
   intro z hz
   simp only [splits_X_sub_C (RingHom.id K)]
 #align polynomial.cyclotomic'_splits Polynomial.cyclotomic'_splits
+-/
 
 /-- If there is a primitive `n`-th root of unity in `K`, then `X ^ n - 1`splits. -/
-theorem x_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
+theorem X_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
     Splits (RingHom.id K) (X ^ n - C (1 : K)) := by
   rw [splits_iff_card_roots, ← nth_roots, IsPrimitiveRoot.card_nthRoots h, nat_degree_X_pow_sub_C]
-#align polynomial.X_pow_sub_one_splits Polynomial.x_pow_sub_one_splits
+#align polynomial.X_pow_sub_one_splits Polynomial.X_pow_sub_one_splits
 
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `∏ i in nat.divisors n, cyclotomic' i K = X ^ n - 1`. -/
-theorem prod_cyclotomic'_eq_x_pow_sub_one {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
+theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     (∏ i in Nat.divisors n, cyclotomic' i K) = X ^ n - 1 :=
   by
@@ -181,11 +199,11 @@ theorem prod_cyclotomic'_eq_x_pow_sub_one {K : Type _} [CommRing K] [IsDomain K]
     fun x hx y hy hne => IsPrimitiveRoot.disjoint hne
   simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← Finset.prod_biUnion hd,
     h.nth_roots_one_eq_bUnion_primitive_roots]
-#align polynomial.prod_cyclotomic'_eq_X_pow_sub_one Polynomial.prod_cyclotomic'_eq_x_pow_sub_one
+#align polynomial.prod_cyclotomic'_eq_X_pow_sub_one Polynomial.prod_cyclotomic'_eq_X_pow_sub_one
 
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `cyclotomic' n K = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic' i K)`. -/
-theorem cyclotomic'_eq_x_pow_sub_one_div {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
+theorem cyclotomic'_eq_X_pow_sub_one_div {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     cyclotomic' n K = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic' i K :=
   by
@@ -202,7 +220,7 @@ theorem cyclotomic'_eq_x_pow_sub_one_div {K : Type _} [CommRing K] [IsDomain K]
   rw [bot_lt_iff_ne_bot]
   intro h
   exact monic.ne_zero prod_monic (degree_eq_bot.1 h)
-#align polynomial.cyclotomic'_eq_X_pow_sub_one_div Polynomial.cyclotomic'_eq_x_pow_sub_one_div
+#align polynomial.cyclotomic'_eq_X_pow_sub_one_div Polynomial.cyclotomic'_eq_X_pow_sub_one_div
 
 /-- If there is a primitive `n`-th root of unity in `K`, then `cyclotomic' n K` comes from a
 monic polynomial with integer coefficients. -/
@@ -263,11 +281,13 @@ end Cyclotomic'
 
 section Cyclotomic
 
+#print Polynomial.cyclotomic /-
 /-- The `n`-th cyclotomic polynomial with coefficients in `R`. -/
 def cyclotomic (n : ℕ) (R : Type _) [Ring R] : R[X] :=
   if h : n = 0 then 1
   else map (Int.castRingHom R) (int_coeff_of_cyclotomic' (Complex.isPrimitiveRoot_exp n h)).some
 #align polynomial.cyclotomic Polynomial.cyclotomic
+-/
 
 theorem int_cyclotomic_rw {n : ℕ} (h : n ≠ 0) :
     cyclotomic n ℤ = (int_coeff_of_cyclotomic' (Complex.isPrimitiveRoot_exp n h)).some :=
@@ -319,12 +339,15 @@ theorem cyclotomic.eval_apply {R S : Type _} (q : R) (n : ℕ) [Ring R] [Ring S]
   rw [← map_cyclotomic n f, eval_map, eval₂_at_apply]
 #align polynomial.cyclotomic.eval_apply Polynomial.cyclotomic.eval_apply
 
+#print Polynomial.cyclotomic_zero /-
 /-- The zeroth cyclotomic polyomial is `1`. -/
 @[simp]
 theorem cyclotomic_zero (R : Type _) [Ring R] : cyclotomic 0 R = 1 := by
   simp only [cyclotomic, dif_pos]
 #align polynomial.cyclotomic_zero Polynomial.cyclotomic_zero
+-/
 
+#print Polynomial.cyclotomic_one /-
 /-- The first cyclotomic polyomial is `X - 1`. -/
 @[simp]
 theorem cyclotomic_one (R : Type _) [Ring R] : cyclotomic 1 R = X - 1 :=
@@ -335,24 +358,32 @@ theorem cyclotomic_one (R : Type _) [Ring R] : cyclotomic 1 R = X - 1 :=
   rw [← map_cyclotomic_int, ← int_cyclotomic_unique hspec]
   simp only [map_X, Polynomial.map_one, Polynomial.map_sub]
 #align polynomial.cyclotomic_one Polynomial.cyclotomic_one
+-/
 
+#print Polynomial.cyclotomic.monic /-
 /-- `cyclotomic n` is monic. -/
 theorem cyclotomic.monic (n : ℕ) (R : Type _) [Ring R] : (cyclotomic n R).Monic :=
   by
   rw [← map_cyclotomic_int]
   exact (int_cyclotomic_spec n).2.2.map _
 #align polynomial.cyclotomic.monic Polynomial.cyclotomic.monic
+-/
 
+#print Polynomial.cyclotomic.isPrimitive /-
 /-- `cyclotomic n` is primitive. -/
 theorem cyclotomic.isPrimitive (n : ℕ) (R : Type _) [CommRing R] : (cyclotomic n R).IsPrimitive :=
   (cyclotomic.monic n R).IsPrimitive
 #align polynomial.cyclotomic.is_primitive Polynomial.cyclotomic.isPrimitive
+-/
 
+#print Polynomial.cyclotomic_ne_zero /-
 /-- `cyclotomic n R` is different from `0`. -/
 theorem cyclotomic_ne_zero (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] : cyclotomic n R ≠ 0 :=
   (cyclotomic.monic n R).NeZero
 #align polynomial.cyclotomic_ne_zero Polynomial.cyclotomic_ne_zero
+-/
 
+#print Polynomial.degree_cyclotomic /-
 /-- The degree of `cyclotomic n` is `totient n`. -/
 theorem degree_cyclotomic (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] :
     (cyclotomic n R).degree = Nat.totient n :=
@@ -366,23 +397,28 @@ theorem degree_cyclotomic (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] :
   simp only [(int_cyclotomic_spec n).right.right, eq_intCast, monic.leading_coeff, Int.cast_one,
     Ne.def, not_false_iff, one_ne_zero]
 #align polynomial.degree_cyclotomic Polynomial.degree_cyclotomic
+-/
 
+#print Polynomial.natDegree_cyclotomic /-
 /-- The natural degree of `cyclotomic n` is `totient n`. -/
 theorem natDegree_cyclotomic (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] :
     (cyclotomic n R).natDegree = Nat.totient n := by
   rw [nat_degree, degree_cyclotomic, WithBot.unbot'_coe]
 #align polynomial.nat_degree_cyclotomic Polynomial.natDegree_cyclotomic
+-/
 
+#print Polynomial.degree_cyclotomic_pos /-
 /-- The degree of `cyclotomic n R` is positive. -/
 theorem degree_cyclotomic_pos (n : ℕ) (R : Type _) (hpos : 0 < n) [Ring R] [Nontrivial R] :
     0 < (cyclotomic n R).degree := by rw [degree_cyclotomic n R];
   exact_mod_cast Nat.totient_pos hpos
 #align polynomial.degree_cyclotomic_pos Polynomial.degree_cyclotomic_pos
+-/
 
 open Finset
 
 /-- `∏ i in nat.divisors n, cyclotomic i R = X ^ n - 1`. -/
-theorem prod_cyclotomic_eq_x_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [CommRing R] :
+theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [CommRing R] :
     (∏ i in Nat.divisors n, cyclotomic i R) = X ^ n - 1 :=
   by
   have integer : (∏ i in Nat.divisors n, cyclotomic i ℤ) = X ^ n - 1 :=
@@ -393,9 +429,9 @@ theorem prod_cyclotomic_eq_x_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [
     exact prod_cyclotomic'_eq_X_pow_sub_one hpos (Complex.isPrimitiveRoot_exp n hpos.ne')
   simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one,
     Polynomial.map_pow, Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) integer
-#align polynomial.prod_cyclotomic_eq_X_pow_sub_one Polynomial.prod_cyclotomic_eq_x_pow_sub_one
+#align polynomial.prod_cyclotomic_eq_X_pow_sub_one Polynomial.prod_cyclotomic_eq_X_pow_sub_one
 
-theorem cyclotomic.dvd_x_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomic n R ∣ X ^ n - 1 :=
+theorem cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomic n R ∣ X ^ n - 1 :=
   by
   suffices cyclotomic n ℤ ∣ X ^ n - 1 by
     simpa only [map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
@@ -404,7 +440,7 @@ theorem cyclotomic.dvd_x_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomi
   · simp
   rw [← prod_cyclotomic_eq_X_pow_sub_one hn]
   exact Finset.dvd_prod_of_mem _ (n.mem_divisors_self hn.ne')
-#align polynomial.cyclotomic.dvd_X_pow_sub_one Polynomial.cyclotomic.dvd_x_pow_sub_one
+#align polynomial.cyclotomic.dvd_X_pow_sub_one Polynomial.cyclotomic.dvd_X_pow_sub_one
 
 theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
     (∏ i in n.divisors.eraseₓ 1, cyclotomic i R) = ∑ i in Finset.range n, X ^ i :=
@@ -427,13 +463,15 @@ theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
     erase_insert (mem_singleton.not.2 hp.out.ne_one.symm), prod_singleton]
 #align polynomial.cyclotomic_prime Polynomial.cyclotomic_prime
 
-theorem cyclotomic_prime_mul_x_sub_one (R : Type _) [Ring R] (p : ℕ) [hn : Fact (Nat.Prime p)] :
+theorem cyclotomic_prime_mul_X_sub_one (R : Type _) [Ring R] (p : ℕ) [hn : Fact (Nat.Prime p)] :
     cyclotomic p R * (X - 1) = X ^ p - 1 := by rw [cyclotomic_prime, geom_sum_mul]
-#align polynomial.cyclotomic_prime_mul_X_sub_one Polynomial.cyclotomic_prime_mul_x_sub_one
+#align polynomial.cyclotomic_prime_mul_X_sub_one Polynomial.cyclotomic_prime_mul_X_sub_one
 
+#print Polynomial.cyclotomic_two /-
 @[simp]
 theorem cyclotomic_two (R : Type _) [Ring R] : cyclotomic 2 R = X + 1 := by simp [cyclotomic_prime]
 #align polynomial.cyclotomic_two Polynomial.cyclotomic_two
+-/
 
 @[simp]
 theorem cyclotomic_three (R : Type _) [Ring R] : cyclotomic 3 R = X ^ 2 + X + 1 := by
@@ -453,7 +491,7 @@ theorem cyclotomic_dvd_geom_sum_of_dvd (R) [Ring R] {d n : ℕ} (hdn : d ∣ n)
   simp [hd, hdn, hn.ne']
 #align polynomial.cyclotomic_dvd_geom_sum_of_dvd Polynomial.cyclotomic_dvd_geom_sum_of_dvd
 
-theorem x_pow_sub_one_mul_prod_cyclotomic_eq_x_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
+theorem X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
     (h : d ∈ n.properDivisors) :
     ((X ^ d - 1) * ∏ x in n.divisors \ d.divisors, cyclotomic x R) = X ^ n - 1 :=
   by
@@ -462,9 +500,9 @@ theorem x_pow_sub_one_mul_prod_cyclotomic_eq_x_pow_sub_one_of_dvd (R) [CommRing
   have h0d : 0 < d := Nat.pos_of_dvd_of_pos hd h0n
   rw [← prod_cyclotomic_eq_X_pow_sub_one h0d, ← prod_cyclotomic_eq_X_pow_sub_one h0n, mul_comm,
     Finset.prod_sdiff (Nat.divisors_subset_of_dvd h0n.ne' hd)]
-#align polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd Polynomial.x_pow_sub_one_mul_prod_cyclotomic_eq_x_pow_sub_one_of_dvd
+#align polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd Polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd
 
-theorem x_pow_sub_one_mul_cyclotomic_dvd_x_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
+theorem X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
     (h : d ∈ n.properDivisors) : (X ^ d - 1) * cyclotomic n R ∣ X ^ n - 1 :=
   by
   have hdn := (nat.mem_proper_divisors.mp h).2
@@ -476,7 +514,7 @@ theorem x_pow_sub_one_mul_cyclotomic_dvd_x_pow_sub_one_of_dvd (R) [CommRing R] {
   rw [← Nat.insert_self_properDivisors hdn.ne_bot, insert_sdiff_of_not_mem, prod_insert]
   · exact Finset.not_mem_sdiff_of_not_mem_left Nat.properDivisors.not_self_mem
   · exact fun hk => hdn.not_le <| Nat.divisor_le hk
-#align polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd Polynomial.x_pow_sub_one_mul_cyclotomic_dvd_x_pow_sub_one_of_dvd
+#align polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd Polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd
 
 section ArithmeticFunction
 
@@ -486,7 +524,7 @@ open scoped ArithmeticFunction
 
 /-- `cyclotomic n R` can be expressed as a product in a fraction field of `R[X]`
   using Möbius inversion. -/
-theorem cyclotomic_eq_prod_x_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [CommRing R]
+theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [CommRing R]
     [IsDomain R] :
     algebraMap _ (RatFunc R) (cyclotomic n R) =
       ∏ i in n.divisorsAntidiagonal, algebraMap R[X] _ (X ^ i.snd - 1) ^ μ i.fst :=
@@ -506,13 +544,13 @@ theorem cyclotomic_eq_prod_x_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [Com
   · apply cyclotomic_ne_zero
   · apply monic.ne_zero
     apply monic_X_pow_sub_C _ (ne_of_gt hn)
-#align polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius Polynomial.cyclotomic_eq_prod_x_pow_sub_one_pow_moebius
+#align polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius
 
 end ArithmeticFunction
 
 /-- We have
 `cyclotomic n R = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic i K)`. -/
-theorem cyclotomic_eq_x_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) :
+theorem cyclotomic_eq_X_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) :
     cyclotomic n R = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R :=
   by
   nontriviality R
@@ -530,11 +568,11 @@ theorem cyclotomic_eq_x_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpo
   rw [bot_lt_iff_ne_bot]
   intro h
   exact monic.ne_zero prod_monic (degree_eq_bot.1 h)
-#align polynomial.cyclotomic_eq_X_pow_sub_one_div Polynomial.cyclotomic_eq_x_pow_sub_one_div
+#align polynomial.cyclotomic_eq_X_pow_sub_one_div Polynomial.cyclotomic_eq_X_pow_sub_one_div
 
 /-- If `m` is a proper divisor of `n`, then `X ^ m - 1` divides
 `∏ i in nat.proper_divisors n, cyclotomic i R`. -/
-theorem x_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ} (hpos : 0 < n)
+theorem X_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ} (hpos : 0 < n)
     (hm : m ∣ n) (hdiff : m ≠ n) : X ^ m - 1 ∣ ∏ i in Nat.properDivisors n, cyclotomic i R :=
   by
   replace hm :=
@@ -547,12 +585,12 @@ theorem x_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ}
     Finset.prod_union Finset.sdiff_disjoint,
     prod_cyclotomic_eq_X_pow_sub_one (Nat.pos_of_mem_properDivisors hm)]
   exact ⟨∏ x : ℕ in n.proper_divisors \ m.divisors, cyclotomic x R, by rw [mul_comm]⟩
-#align polynomial.X_pow_sub_one_dvd_prod_cyclotomic Polynomial.x_pow_sub_one_dvd_prod_cyclotomic
+#align polynomial.X_pow_sub_one_dvd_prod_cyclotomic Polynomial.X_pow_sub_one_dvd_prod_cyclotomic
 
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `cyclotomic n K = ∏ μ in primitive_roots n R, (X - C μ)`. In particular,
 `cyclotomic n K = cyclotomic' n K` -/
-theorem cyclotomic_eq_prod_x_sub_primitiveRoots {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
+theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
     {n : ℕ} (hz : IsPrimitiveRoot ζ n) : cyclotomic n K = ∏ μ in primitiveRoots n K, X - C μ :=
   by
   rw [← cyclotomic']
@@ -567,7 +605,7 @@ theorem cyclotomic_eq_prod_x_sub_primitiveRoots {K : Type _} [CommRing K] [IsDom
     exact hk i (Nat.mem_properDivisors.1 hi).2 (IsPrimitiveRoot.pow hpos hz hd)
   rw [@cyclotomic_eq_X_pow_sub_one_div _ _ _ hpos, cyclotomic'_eq_X_pow_sub_one_div hpos hz,
     Finset.prod_congr (refl k.proper_divisors) h]
-#align polynomial.cyclotomic_eq_prod_X_sub_primitive_roots Polynomial.cyclotomic_eq_prod_x_sub_primitiveRoots
+#align polynomial.cyclotomic_eq_prod_X_sub_primitive_roots Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots
 
 theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P : R[X]) :
     P = cyclotomic n R ↔ (P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R) = X ^ n - 1 :=
@@ -613,11 +651,11 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
   rw [mul_comm _ (∑ i in _, _), n_ih, geom_sum_mul, sub_left_inj, ← pow_mul, pow_add, pow_one]
 #align polynomial.cyclotomic_prime_pow_eq_geom_sum Polynomial.cyclotomic_prime_pow_eq_geom_sum
 
-theorem cyclotomic_prime_pow_mul_x_pow_sub_one (R : Type _) [CommRing R] (p k : ℕ)
+theorem cyclotomic_prime_pow_mul_X_pow_sub_one (R : Type _) [CommRing R] (p k : ℕ)
     [hn : Fact (Nat.Prime p)] :
     cyclotomic (p ^ (k + 1)) R * (X ^ p ^ k - 1) = X ^ p ^ (k + 1) - 1 := by
   rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, ← pow_mul, pow_succ, mul_comm]
-#align polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one Polynomial.cyclotomic_prime_pow_mul_x_pow_sub_one
+#align polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one
 
 /-- The constant term of `cyclotomic n R` is `1` if `2 ≤ n`. -/
 theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -4,20 +4,20 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.cyclotomic.basic
-! leanprover-community/mathlib commit 5bfbcca0a7ffdd21cf1682e59106d6c942434a32
+! leanprover-community/mathlib commit 7fdeecc0d03cd40f7a165e6cf00a4d2286db599f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.NeZero
 import Mathbin.Algebra.Polynomial.BigOperators
-import Mathbin.Analysis.Complex.RootsOfUnity
+import Mathbin.RingTheory.RootsOfUnity.Complex
 import Mathbin.Data.Polynomial.Lifts
 import Mathbin.Data.Polynomial.Splits
 import Mathbin.Data.Zmod.Algebra
 import Mathbin.FieldTheory.Ratfunc
 import Mathbin.FieldTheory.Separable
 import Mathbin.NumberTheory.ArithmeticFunction
-import Mathbin.RingTheory.RootsOfUnity
+import Mathbin.RingTheory.RootsOfUnity.Basic
 
 /-!
 # Cyclotomic polynomials.

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -227,7 +227,7 @@ theorem int_coeff_of_cyclotomic' {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
     intro x hx
     have xsmall := (Nat.mem_properDivisors.1 hx).2
     obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hx).1
-    rw [mul_comm] at hd
+    rw [mul_comm] at hd 
     exact ihk x xsmall (h.pow hpos hd)
   replace Bint := lifts_and_degree_eq_and_monic Bint Bmo
   obtain ⟨B₁, hB₁, hB₁deg, hB₁mo⟩ := Bint
@@ -563,7 +563,7 @@ theorem cyclotomic_eq_prod_x_sub_primitiveRoots {K : Type _} [CommRing K] [IsDom
     by
     intro i hi
     obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hi).1
-    rw [mul_comm] at hd
+    rw [mul_comm] at hd 
     exact hk i (Nat.mem_properDivisors.1 hi).2 (IsPrimitiveRoot.pow hpos hz hd)
   rw [@cyclotomic_eq_X_pow_sub_one_div _ _ _ hpos, cyclotomic'_eq_X_pow_sub_one_div hpos hz,
     Finset.prod_congr (refl k.proper_divisors) h]
@@ -603,13 +603,13 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
     by
     intro m
     have := eq_cyclotomic_iff (pow_pos hp.pos (m + 1)) _
-    rw [eq_comm] at this
+    rw [eq_comm] at this 
     rw [this, Nat.prod_properDivisors_prime_pow hp]
   induction' n with n_n n_ih
   · haveI := Fact.mk hp; simp [cyclotomic_prime]
   rw [((eq_cyclotomic_iff (pow_pos hp.pos (n_n.succ + 1)) _).mpr _).symm]
   rw [Nat.prod_properDivisors_prime_pow hp, Finset.prod_range_succ, n_ih]
-  rw [this] at n_ih
+  rw [this] at n_ih 
   rw [mul_comm _ (∑ i in _, _), n_ih, geom_sum_mul, sub_left_inj, ← pow_mul, pow_add, pow_one]
 #align polynomial.cyclotomic_prime_pow_eq_geom_sum Polynomial.cyclotomic_prime_pow_eq_geom_sum
 
@@ -654,7 +654,7 @@ theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
     rw [coeff_zero_eq_eval_zero _]
     simp only [zero_pow (lt_of_lt_of_le zero_lt_two hn), eval_X, eval_one, zero_sub, eval_pow,
       eval_sub]
-  rw [hzero] at heq
+  rw [hzero] at heq 
   exact neg_inj.mp (Eq.symm HEq)
 #align polynomial.cyclotomic_coeff_zero Polynomial.cyclotomic_coeff_zero
 
@@ -667,14 +667,14 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h
-  rw [is_root.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
+  rw [is_root.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot 
   by_cases hone : n = 1
   · simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
-      one_ne_zero, coeff_sub] at hroot
+      one_ne_zero, coeff_sub] at hroot 
     exact hroot
   rw [cyclotomic_coeff_zero (ZMod p)
       (Nat.succ_le_of_lt (lt_of_le_of_ne (Nat.succ_le_of_lt hpos) (Ne.symm hone)))] at
-    hroot
+    hroot 
   exact one_ne_zero hroot
 #align polynomial.coprime_of_root_cyclotomic Polynomial.coprime_of_root_cyclotomic
 
@@ -690,10 +690,10 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
   by
   apply orderOf_dvd_of_pow_eq_one
   suffices hpow : eval (Nat.castRingHom (ZMod p) a) (X ^ n - 1 : (ZMod p)[X]) = 0
-  · simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
+  · simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow 
     apply Units.val_eq_one.1
     simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
-  rw [is_root.def] at hroot
+  rw [is_root.def] at hroot 
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
     Finset.prod_cons, eval_mul, hroot, MulZeroClass.zero_mul]
 #align polynomial.order_of_root_cyclotomic_dvd Polynomial.orderOf_root_cyclotomic_dvd

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -55,7 +55,7 @@ of unity `μ : K`, where `K` is a field of characteristic `0`.
 -/
 
 
-open Classical BigOperators Polynomial
+open scoped Classical BigOperators Polynomial
 
 noncomputable section
 
@@ -482,7 +482,7 @@ section ArithmeticFunction
 
 open Nat.ArithmeticFunction
 
-open ArithmeticFunction
+open scoped ArithmeticFunction
 
 /-- `cyclotomic n R` can be expressed as a product in a fraction field of `R[X]`
   using Möbius inversion. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -132,9 +132,7 @@ theorem degree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
 
 /-- The roots of `cyclotomic' n R` are the primitive `n`-th roots of unity. -/
 theorem roots_of_cyclotomic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
-    (cyclotomic' n R).roots = (primitiveRoots n R).val :=
-  by
-  rw [cyclotomic']
+    (cyclotomic' n R).roots = (primitiveRoots n R).val := by rw [cyclotomic'];
   exact roots_prod_X_sub_C (primitiveRoots n R)
 #align polynomial.roots_of_cyclotomic Polynomial.roots_of_cyclotomic
 
@@ -377,8 +375,7 @@ theorem natDegree_cyclotomic (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] :
 
 /-- The degree of `cyclotomic n R` is positive. -/
 theorem degree_cyclotomic_pos (n : ℕ) (R : Type _) (hpos : 0 < n) [Ring R] [Nontrivial R] :
-    0 < (cyclotomic n R).degree := by
-  rw [degree_cyclotomic n R]
+    0 < (cyclotomic n R).degree := by rw [degree_cyclotomic n R];
   exact_mod_cast Nat.totient_pos hpos
 #align polynomial.degree_cyclotomic_pos Polynomial.degree_cyclotomic_pos
 
@@ -609,8 +606,7 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
     rw [eq_comm] at this
     rw [this, Nat.prod_properDivisors_prime_pow hp]
   induction' n with n_n n_ih
-  · haveI := Fact.mk hp
-    simp [cyclotomic_prime]
+  · haveI := Fact.mk hp; simp [cyclotomic_prime]
   rw [((eq_cyclotomic_iff (pow_pos hp.pos (n_n.succ + 1)) _).mpr _).symm]
   rw [Nat.prod_properDivisors_prime_pow hp, Finset.prod_range_succ, n_ih]
   rw [this] at n_ih

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -491,7 +491,7 @@ open ArithmeticFunction
   using Möbius inversion. -/
 theorem cyclotomic_eq_prod_x_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [CommRing R]
     [IsDomain R] :
-    algebraMap _ (Ratfunc R) (cyclotomic n R) =
+    algebraMap _ (RatFunc R) (cyclotomic n R) =
       ∏ i in n.divisorsAntidiagonal, algebraMap R[X] _ (X ^ i.snd - 1) ^ μ i.fst :=
   by
   rcases n.eq_zero_or_pos with (rfl | hpos)
@@ -499,7 +499,7 @@ theorem cyclotomic_eq_prod_x_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [Com
   have h :
     ∀ n : ℕ,
       0 < n →
-        (∏ i in Nat.divisors n, algebraMap _ (Ratfunc R) (cyclotomic i R)) =
+        (∏ i in Nat.divisors n, algebraMap _ (RatFunc R) (cyclotomic i R)) =
           algebraMap _ _ (X ^ n - 1) :=
     by
     intro n hn

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.cyclotomic.basic
-! leanprover-community/mathlib commit 2032a878972d5672e7c27c957e7a6e297b044973
+! leanprover-community/mathlib commit 5bfbcca0a7ffdd21cf1682e59106d6c942434a32
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -34,13 +34,11 @@ with coefficients in any ring `R`.
 
 ## Main results
 
-* `int_coeff_of_cycl` : If there is a primitive `n`-th root of unity in `K`, then `cyclotomic' n K`
-comes from a polynomial with integer coefficients.
-* `deg_of_cyclotomic` : The degree of `cyclotomic n` is `totient n`.
-* `prod_cyclotomic_eq_X_pow_sub_one` : `X ^ n - 1 = ∏ (cyclotomic i)`, where `i` divides `n`.
-* `cyclotomic_eq_prod_X_pow_sub_one_pow_moebius` : The Möbius inversion formula for
+* `polynomial.degree_cyclotomic` : The degree of `cyclotomic n` is `totient n`.
+* `polynomial.prod_cyclotomic_eq_X_pow_sub_one` : `X ^ n - 1 = ∏ (cyclotomic i)`, where `i`
+  divides `n`.
+* `polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius` : The Möbius inversion formula for
   `cyclotomic n R` over an abstract fraction field for `R[X]`.
-* `cyclotomic.irreducible` : `cyclotomic n ℤ` is irreducible.
 
 ## Implementation details
 
@@ -483,26 +481,6 @@ theorem x_pow_sub_one_mul_cyclotomic_dvd_x_pow_sub_one_of_dvd (R) [CommRing R] {
   · exact fun hk => hdn.not_le <| Nat.divisor_le hk
 #align polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd Polynomial.x_pow_sub_one_mul_cyclotomic_dvd_x_pow_sub_one_of_dvd
 
-theorem isRoot_of_unity_iff {n : ℕ} (h : 0 < n) (R : Type _) [CommRing R] [IsDomain R] {ζ : R} :
-    ζ ^ n = 1 ↔ ∃ i ∈ n.divisors, (cyclotomic i R).IsRoot ζ := by
-  rw [← mem_nth_roots h, nth_roots, mem_roots <| X_pow_sub_C_ne_zero h _, C_1, ←
-      prod_cyclotomic_eq_X_pow_sub_one h, is_root_prod] <;>
-    infer_instance
-#align is_root_of_unity_iff isRoot_of_unity_iff
-
-theorem isRoot_of_unity_of_root_cyclotomic {n : ℕ} {R} [CommRing R] {ζ : R} {i : ℕ}
-    (hi : i ∈ n.divisors) (h : (cyclotomic i R).IsRoot ζ) : ζ ^ n = 1 :=
-  by
-  rcases n.eq_zero_or_pos with (rfl | hn)
-  · exact pow_zero _
-  have := congr_arg (eval ζ) (prod_cyclotomic_eq_X_pow_sub_one hn R).symm
-  rw [eval_sub, eval_pow, eval_X, eval_one] at this
-  convert eq_add_of_sub_eq' this
-  convert(add_zero _).symm
-  apply eval_eq_zero_of_dvd_of_eval_eq_zero _ h
-  exact Finset.dvd_prod_of_mem _ hi
-#align polynomial.is_root_of_unity_of_root_cyclotomic Polynomial.isRoot_of_unity_of_root_cyclotomic
-
 section ArithmeticFunction
 
 open Nat.ArithmeticFunction
@@ -594,124 +572,6 @@ theorem cyclotomic_eq_prod_x_sub_primitiveRoots {K : Type _} [CommRing K] [IsDom
     Finset.prod_congr (refl k.proper_divisors) h]
 #align polynomial.cyclotomic_eq_prod_X_sub_primitive_roots Polynomial.cyclotomic_eq_prod_x_sub_primitiveRoots
 
-section Roots
-
-variable {R : Type _} {n : ℕ} [CommRing R] [IsDomain R]
-
-/-- Any `n`-th primitive root of unity is a root of `cyclotomic n K`.-/
-theorem IsPrimitiveRoot.isRoot_cyclotomic (hpos : 0 < n) {μ : R} (h : IsPrimitiveRoot μ n) :
-    IsRoot (cyclotomic n R) μ :=
-  by
-  rw [← mem_roots (cyclotomic_ne_zero n R), cyclotomic_eq_prod_X_sub_primitive_roots h,
-    roots_prod_X_sub_C, ← Finset.mem_def]
-  rwa [← mem_primitiveRoots hpos] at h
-#align is_primitive_root.is_root_cyclotomic IsPrimitiveRoot.isRoot_cyclotomic
-
-private theorem is_root_cyclotomic_iff' {n : ℕ} {K : Type _} [Field K] {μ : K} [NeZero (n : K)] :
-    IsRoot (cyclotomic n K) μ ↔ IsPrimitiveRoot μ n :=
-  by
-  -- in this proof, `o` stands for `order_of μ`
-  have hnpos : 0 < n := (NeZero.of_neZero_natCast K).out.bot_lt
-  refine' ⟨fun hμ => _, IsPrimitiveRoot.isRoot_cyclotomic hnpos⟩
-  have hμn : μ ^ n = 1 := by
-    rw [isRoot_of_unity_iff hnpos]
-    exact ⟨n, n.mem_divisors_self hnpos.ne', hμ⟩
-  by_contra hnμ
-  have ho : 0 < orderOf μ := by
-    apply orderOf_pos'
-    rw [isOfFinOrder_iff_pow_eq_one]
-    exact ⟨n, hnpos, hμn⟩
-  have := pow_orderOf_eq_one μ
-  rw [isRoot_of_unity_iff ho] at this
-  obtain ⟨i, hio, hiμ⟩ := this
-  replace hio := Nat.dvd_of_mem_divisors hio
-  rw [IsPrimitiveRoot.not_iff] at hnμ
-  rw [← orderOf_dvd_iff_pow_eq_one] at hμn
-  have key : i < n := (Nat.le_of_dvd ho hio).trans_lt ((Nat.le_of_dvd hnpos hμn).lt_of_ne hnμ)
-  have key' : i ∣ n := hio.trans hμn
-  rw [← Polynomial.dvd_iff_isRoot] at hμ hiμ
-  have hni : {i, n} ⊆ n.divisors := by simpa [Finset.insert_subset, key'] using hnpos.ne'
-  obtain ⟨k, hk⟩ := hiμ
-  obtain ⟨j, hj⟩ := hμ
-  have := prod_cyclotomic_eq_X_pow_sub_one hnpos K
-  rw [← Finset.prod_sdiff hni, Finset.prod_pair key.ne, hk, hj] at this
-  have hn := (X_pow_sub_one_separable_iff.mpr <| NeZero.natCast_ne n K).Squarefree
-  rw [← this, Squarefree] at hn
-  contrapose! hn
-  refine' ⟨X - C μ, ⟨(∏ x in n.divisors \ {i, n}, cyclotomic x K) * k * j, by ring⟩, _⟩
-  simp [Polynomial.isUnit_iff_degree_eq_zero]
-#align polynomial.is_root_cyclotomic_iff' polynomial.is_root_cyclotomic_iff'
-
-theorem isRoot_cyclotomic_iff [NeZero (n : R)] {μ : R} :
-    IsRoot (cyclotomic n R) μ ↔ IsPrimitiveRoot μ n :=
-  by
-  have hf : Function.Injective _ := IsFractionRing.injective R (FractionRing R)
-  haveI : NeZero (n : FractionRing R) := NeZero.nat_of_injective hf
-  rw [← is_root_map_iff hf, ← IsPrimitiveRoot.map_iff_of_injective hf, map_cyclotomic, ←
-    is_root_cyclotomic_iff']
-#align polynomial.is_root_cyclotomic_iff Polynomial.isRoot_cyclotomic_iff
-
-theorem roots_cyclotomic_nodup [NeZero (n : R)] : (cyclotomic n R).roots.Nodup :=
-  by
-  obtain h | ⟨ζ, hζ⟩ := (cyclotomic n R).roots.empty_or_exists_mem
-  · exact h.symm ▸ Multiset.nodup_zero
-  rw [mem_roots <| cyclotomic_ne_zero n R, is_root_cyclotomic_iff] at hζ
-  refine'
-    Multiset.nodup_of_le
-      (roots.le_of_dvd (X_pow_sub_C_ne_zero (NeZero.pos_of_neZero_natCast R) 1) <|
-        cyclotomic.dvd_X_pow_sub_one n R)
-      hζ.nth_roots_nodup
-#align polynomial.roots_cyclotomic_nodup Polynomial.roots_cyclotomic_nodup
-
-theorem cyclotomic.roots_to_finset_eq_primitiveRoots [NeZero (n : R)] :
-    (⟨(cyclotomic n R).roots, roots_cyclotomic_nodup⟩ : Finset _) = primitiveRoots n R :=
-  by
-  ext
-  simp [cyclotomic_ne_zero n R, is_root_cyclotomic_iff, mem_primitiveRoots,
-    NeZero.pos_of_neZero_natCast R]
-#align polynomial.cyclotomic.roots_to_finset_eq_primitive_roots Polynomial.cyclotomic.roots_to_finset_eq_primitiveRoots
-
-theorem cyclotomic.roots_eq_primitiveRoots_val [NeZero (n : R)] :
-    (cyclotomic n R).roots = (primitiveRoots n R).val := by
-  rw [← cyclotomic.roots_to_finset_eq_primitive_roots]
-#align polynomial.cyclotomic.roots_eq_primitive_roots_val Polynomial.cyclotomic.roots_eq_primitiveRoots_val
-
-end Roots
-
-/-- If `R` is of characteristic zero, then `ζ` is a root of `cyclotomic n R` if and only if it is a
-primitive `n`-th root of unity. -/
-theorem isRoot_cyclotomic_iff_charZero {n : ℕ} {R : Type _} [CommRing R] [IsDomain R] [CharZero R]
-    {μ : R} (hn : 0 < n) : (Polynomial.cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n :=
-  letI := NeZero.of_gt hn
-  is_root_cyclotomic_iff
-#align polynomial.is_root_cyclotomic_iff_char_zero Polynomial.isRoot_cyclotomic_iff_charZero
-
-/-- Over a ring `R` of characteristic zero, `λ n, cyclotomic n R` is injective. -/
-theorem cyclotomic_injective {R : Type _} [CommRing R] [CharZero R] :
-    Function.Injective fun n => cyclotomic n R :=
-  by
-  intro n m hnm
-  simp only at hnm
-  rcases eq_or_ne n 0 with (rfl | hzero)
-  · rw [cyclotomic_zero] at hnm
-    replace hnm := congr_arg nat_degree hnm
-    rw [nat_degree_one, nat_degree_cyclotomic] at hnm
-    by_contra
-    exact (Nat.totient_pos (zero_lt_iff.2 (Ne.symm h))).Ne hnm
-  · haveI := NeZero.mk hzero
-    rw [← map_cyclotomic_int _ R, ← map_cyclotomic_int _ R] at hnm
-    replace hnm := map_injective (Int.castRingHom R) Int.cast_injective hnm
-    replace hnm := congr_arg (map (Int.castRingHom ℂ)) hnm
-    rw [map_cyclotomic_int, map_cyclotomic_int] at hnm
-    have hprim := Complex.isPrimitiveRoot_exp _ hzero
-    have hroot := is_root_cyclotomic_iff.2 hprim
-    rw [hnm] at hroot
-    haveI hmzero : NeZero m := ⟨fun h => by simpa [h] using hroot⟩
-    rw [is_root_cyclotomic_iff] at hroot
-    replace hprim := hprim.eq_order_of
-    rwa [← IsPrimitiveRoot.eq_orderOf hroot] at hprim
-#align polynomial.cyclotomic_injective Polynomial.cyclotomic_injective
-
 theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P : R[X]) :
     P = cyclotomic n R ↔ (P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R) = X ^ n - 1 :=
   by
@@ -844,245 +704,5 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
 
 end Order
 
-section minpoly
-
-open IsPrimitiveRoot Complex
-
-/-- The minimal polynomial of a primitive `n`-th root of unity `μ` divides `cyclotomic n ℤ`. -/
-theorem IsPrimitiveRoot.minpoly_dvd_cyclotomic {n : ℕ} {K : Type _} [Field K] {μ : K}
-    (h : IsPrimitiveRoot μ n) (hpos : 0 < n) [CharZero K] : minpoly ℤ μ ∣ cyclotomic n ℤ :=
-  by
-  apply minpoly.isIntegrallyClosed_dvd (IsIntegral h hpos)
-  simpa [aeval_def, eval₂_eq_eval_map, is_root.def] using is_root_cyclotomic hpos h
-#align is_primitive_root.minpoly_dvd_cyclotomic IsPrimitiveRoot.minpoly_dvd_cyclotomic
-
-theorem IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible {K : Type _} [Field K] {R : Type _}
-    [CommRing R] [IsDomain R] {μ : R} {n : ℕ} [Algebra K R] (hμ : IsPrimitiveRoot μ n)
-    (h : Irreducible <| cyclotomic n K) [NeZero (n : K)] : cyclotomic n K = minpoly K μ :=
-  by
-  haveI := NeZero.of_noZeroSMulDivisors K R n
-  refine' minpoly.eq_of_irreducible_of_monic h _ (cyclotomic.monic n K)
-  rwa [aeval_def, eval₂_eq_eval_map, map_cyclotomic, ← is_root.def, is_root_cyclotomic_iff]
-#align is_primitive_root.minpoly_eq_cyclotomic_of_irreducible IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible
-
-/-- `cyclotomic n ℤ` is the minimal polynomial of a primitive `n`-th root of unity `μ`. -/
-theorem cyclotomic_eq_minpoly {n : ℕ} {K : Type _} [Field K] {μ : K} (h : IsPrimitiveRoot μ n)
-    (hpos : 0 < n) [CharZero K] : cyclotomic n ℤ = minpoly ℤ μ :=
-  by
-  refine'
-    eq_of_monic_of_dvd_of_nat_degree_le (minpoly.monic (IsIntegral h hpos)) (cyclotomic.monic n ℤ)
-      (h.minpoly_dvd_cyclotomic hpos) _
-  simpa [nat_degree_cyclotomic n ℤ] using totient_le_degree_minpoly h
-#align polynomial.cyclotomic_eq_minpoly Polynomial.cyclotomic_eq_minpoly
-
-/-- `cyclotomic n ℚ` is the minimal polynomial of a primitive `n`-th root of unity `μ`. -/
-theorem cyclotomic_eq_minpoly_rat {n : ℕ} {K : Type _} [Field K] {μ : K} (h : IsPrimitiveRoot μ n)
-    (hpos : 0 < n) [CharZero K] : cyclotomic n ℚ = minpoly ℚ μ :=
-  by
-  rw [← map_cyclotomic_int, cyclotomic_eq_minpoly h hpos]
-  exact (minpoly.isIntegrallyClosed_eq_field_fractions' _ (IsIntegral h hpos)).symm
-#align polynomial.cyclotomic_eq_minpoly_rat Polynomial.cyclotomic_eq_minpoly_rat
-
-/-- `cyclotomic n ℤ` is irreducible. -/
-theorem cyclotomic.irreducible {n : ℕ} (hpos : 0 < n) : Irreducible (cyclotomic n ℤ) :=
-  by
-  rw [cyclotomic_eq_minpoly (is_primitive_root_exp n hpos.ne') hpos]
-  apply minpoly.irreducible
-  exact (is_primitive_root_exp n hpos.ne').IsIntegral hpos
-#align polynomial.cyclotomic.irreducible Polynomial.cyclotomic.irreducible
-
-/-- `cyclotomic n ℚ` is irreducible. -/
-theorem cyclotomic.irreducible_rat {n : ℕ} (hpos : 0 < n) : Irreducible (cyclotomic n ℚ) :=
-  by
-  rw [← map_cyclotomic_int]
-  exact
-    (is_primitive.irreducible_iff_irreducible_map_fraction_map (cyclotomic.is_primitive n ℤ)).1
-      (cyclotomic.irreducible hpos)
-#align polynomial.cyclotomic.irreducible_rat Polynomial.cyclotomic.irreducible_rat
-
-/-- If `n ≠ m`, then `(cyclotomic n ℚ)` and `(cyclotomic m ℚ)` are coprime. -/
-theorem cyclotomic.isCoprime_rat {n m : ℕ} (h : n ≠ m) :
-    IsCoprime (cyclotomic n ℚ) (cyclotomic m ℚ) :=
-  by
-  rcases n.eq_zero_or_pos with (rfl | hnzero)
-  · exact isCoprime_one_left
-  rcases m.eq_zero_or_pos with (rfl | hmzero)
-  · exact isCoprime_one_right
-  rw [Irreducible.coprime_iff_not_dvd <| cyclotomic.irreducible_rat <| hnzero]
-  exact fun hdiv =>
-    h <|
-      cyclotomic_injective <|
-        eq_of_monic_of_associated (cyclotomic.monic n ℚ) (cyclotomic.monic m ℚ) <|
-          Irreducible.associated_of_dvd (cyclotomic.irreducible_rat hnzero)
-            (cyclotomic.irreducible_rat hmzero) hdiv
-#align polynomial.cyclotomic.is_coprime_rat Polynomial.cyclotomic.isCoprime_rat
-
-end minpoly
-
-section Expand
-
-/-- If `p` is a prime such that `¬ p ∣ n`, then
-`expand R p (cyclotomic n R) = (cyclotomic (n * p) R) * (cyclotomic n R)`. -/
-@[simp]
-theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Nat.Prime p) (hdiv : ¬p ∣ n)
-    (R : Type _) [CommRing R] :
-    expand R p (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R :=
-  by
-  rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
-  · simp
-  haveI := NeZero.of_pos hnpos
-  suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ by
-    rw [← map_cyclotomic_int, ← map_expand, this, Polynomial.map_mul, map_cyclotomic_int]
-  refine'
-    eq_of_monic_of_dvd_of_nat_degree_le ((cyclotomic.monic _ _).mul (cyclotomic.monic _ _))
-      ((cyclotomic.monic n ℤ).expand hp.pos) _ _
-  · refine'
-      (is_primitive.int.dvd_iff_map_cast_dvd_map_cast _ _
-            (is_primitive.mul (cyclotomic.is_primitive (n * p) ℤ) (cyclotomic.is_primitive n ℤ))
-            ((cyclotomic.monic n ℤ).expand hp.pos).IsPrimitive).2
-        _
-    rw [Polynomial.map_mul, map_cyclotomic_int, map_cyclotomic_int, map_expand, map_cyclotomic_int]
-    refine' IsCoprime.mul_dvd (cyclotomic.is_coprime_rat fun h => _) _ _
-    · replace h : n * p = n * 1 := by simp [h]
-      exact Nat.Prime.ne_one hp (mul_left_cancel₀ hnpos.ne' h)
-    · have hpos : 0 < n * p := mul_pos hnpos hp.pos
-      have hprim := Complex.isPrimitiveRoot_exp _ hpos.ne'
-      rw [cyclotomic_eq_minpoly_rat hprim hpos]
-      refine' @minpoly.dvd ℚ ℂ _ _ algebraRat _ _ _
-      rw [aeval_def, ← eval_map, map_expand, map_cyclotomic, expand_eval, ← is_root.def,
-        is_root_cyclotomic_iff]
-      convert IsPrimitiveRoot.pow_of_dvd hprim hp.ne_zero (dvd_mul_left p n)
-      rw [Nat.mul_div_cancel _ (Nat.Prime.pos hp)]
-    · have hprim := Complex.isPrimitiveRoot_exp _ hnpos.ne.symm
-      rw [cyclotomic_eq_minpoly_rat hprim hnpos]
-      refine' @minpoly.dvd ℚ ℂ _ _ algebraRat _ _ _
-      rw [aeval_def, ← eval_map, map_expand, expand_eval, ← is_root.def, ←
-        cyclotomic_eq_minpoly_rat hprim hnpos, map_cyclotomic, is_root_cyclotomic_iff]
-      exact IsPrimitiveRoot.pow_of_prime hprim hp hdiv
-  ·
-    rw [nat_degree_expand, nat_degree_cyclotomic,
-      nat_degree_mul (cyclotomic_ne_zero _ ℤ) (cyclotomic_ne_zero _ ℤ), nat_degree_cyclotomic,
-      nat_degree_cyclotomic, mul_comm n,
-      Nat.totient_mul ((Nat.Prime.coprime_iff_not_dvd hp).2 hdiv), Nat.totient_prime hp,
-      mul_comm (p - 1), ← Nat.mul_succ, Nat.sub_one, Nat.succ_pred_eq_of_pos hp.pos]
-#align polynomial.cyclotomic_expand_eq_cyclotomic_mul Polynomial.cyclotomic_expand_eq_cyclotomic_mul
-
-/-- If `p` is a prime such that `p ∣ n`, then
-`expand R p (cyclotomic n R) = cyclotomic (p * n) R`. -/
-@[simp]
-theorem cyclotomic_expand_eq_cyclotomic {p n : ℕ} (hp : Nat.Prime p) (hdiv : p ∣ n) (R : Type _)
-    [CommRing R] : expand R p (cyclotomic n R) = cyclotomic (n * p) R :=
-  by
-  rcases n.eq_zero_or_pos with (rfl | hzero)
-  · simp
-  haveI := NeZero.of_pos hzero
-  suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ by
-    rw [← map_cyclotomic_int, ← map_expand, this, map_cyclotomic_int]
-  refine'
-    eq_of_monic_of_dvd_of_nat_degree_le (cyclotomic.monic _ _)
-      ((cyclotomic.monic n ℤ).expand hp.pos) _ _
-  · have hpos := Nat.mul_pos hzero hp.pos
-    have hprim := Complex.isPrimitiveRoot_exp _ hpos.ne.symm
-    rw [cyclotomic_eq_minpoly hprim hpos]
-    refine' minpoly.isIntegrallyClosed_dvd (hprim.is_integral hpos) _
-    rw [aeval_def, ← eval_map, map_expand, map_cyclotomic, expand_eval, ← is_root.def,
-      is_root_cyclotomic_iff]
-    · convert IsPrimitiveRoot.pow_of_dvd hprim hp.ne_zero (dvd_mul_left p n)
-      rw [Nat.mul_div_cancel _ hp.pos]
-  ·
-    rw [nat_degree_expand, nat_degree_cyclotomic, nat_degree_cyclotomic, mul_comm n,
-      Nat.totient_mul_of_prime_of_dvd hp hdiv, mul_comm]
-#align polynomial.cyclotomic_expand_eq_cyclotomic Polynomial.cyclotomic_expand_eq_cyclotomic
-
-/-- If the `p ^ n`th cyclotomic polynomial is irreducible, so is the `p ^ m`th, for `m ≤ n`. -/
-theorem cyclotomic_irreducible_pow_of_irreducible_pow {p : ℕ} (hp : Nat.Prime p) {R} [CommRing R]
-    [IsDomain R] {n m : ℕ} (hmn : m ≤ n) (h : Irreducible (cyclotomic (p ^ n) R)) :
-    Irreducible (cyclotomic (p ^ m) R) :=
-  by
-  rcases m.eq_zero_or_pos with (rfl | hm)
-  · simpa using irreducible_X_sub_C (1 : R)
-  obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hmn
-  induction' k with k hk
-  · simpa using h
-  have : m + k ≠ 0 := (add_pos_of_pos_of_nonneg hm k.zero_le).ne'
-  rw [Nat.add_succ, pow_succ', ← cyclotomic_expand_eq_cyclotomic hp <| dvd_pow_self p this] at h
-  exact hk (by linarith) (of_irreducible_expand hp.ne_zero h)
-#align polynomial.cyclotomic_irreducible_pow_of_irreducible_pow Polynomial.cyclotomic_irreducible_pow_of_irreducible_pow
-
-/-- If `irreducible (cyclotomic (p ^ n) R)` then `irreducible (cyclotomic p R).` -/
-theorem cyclotomic_irreducible_of_irreducible_pow {p : ℕ} (hp : Nat.Prime p) {R} [CommRing R]
-    [IsDomain R] {n : ℕ} (hn : n ≠ 0) (h : Irreducible (cyclotomic (p ^ n) R)) :
-    Irreducible (cyclotomic p R) :=
-  pow_one p ▸ cyclotomic_irreducible_pow_of_irreducible_pow hp hn.bot_lt h
-#align polynomial.cyclotomic_irreducible_of_irreducible_pow Polynomial.cyclotomic_irreducible_of_irreducible_pow
-
-end Expand
-
-section CharP
-
-/-- If `R` is of characteristic `p` and `¬p ∣ n`, then
-`cyclotomic (n * p) R = (cyclotomic n R) ^ (p - 1)`. -/
-theorem cyclotomic_mul_prime_eq_pow_of_not_dvd (R : Type _) {p n : ℕ} [hp : Fact (Nat.Prime p)]
-    [Ring R] [CharP R p] (hn : ¬p ∣ n) : cyclotomic (n * p) R = cyclotomic n R ^ (p - 1) :=
-  by
-  suffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ (p - 1) by
-    rw [← map_cyclotomic _ (algebraMap (ZMod p) R), ← map_cyclotomic _ (algebraMap (ZMod p) R),
-      this, Polynomial.map_pow]
-  apply mul_right_injective₀ (cyclotomic_ne_zero n <| ZMod p)
-  rw [← pow_succ, tsub_add_cancel_of_le hp.out.one_lt.le, mul_comm, ← ZMod.expand_card]
-  nth_rw 3 [← map_cyclotomic_int]
-  rw [← map_expand, cyclotomic_expand_eq_cyclotomic_mul hp.out hn, Polynomial.map_mul,
-    map_cyclotomic, map_cyclotomic]
-#align polynomial.cyclotomic_mul_prime_eq_pow_of_not_dvd Polynomial.cyclotomic_mul_prime_eq_pow_of_not_dvd
-
-/-- If `R` is of characteristic `p` and `p ∣ n`, then
-`cyclotomic (n * p) R = (cyclotomic n R) ^ p`. -/
-theorem cyclotomic_mul_prime_dvd_eq_pow (R : Type _) {p n : ℕ} [hp : Fact (Nat.Prime p)] [Ring R]
-    [CharP R p] (hn : p ∣ n) : cyclotomic (n * p) R = cyclotomic n R ^ p :=
-  by
-  suffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p by
-    rw [← map_cyclotomic _ (algebraMap (ZMod p) R), ← map_cyclotomic _ (algebraMap (ZMod p) R),
-      this, Polynomial.map_pow]
-  rw [← ZMod.expand_card, ← map_cyclotomic_int n, ← map_expand,
-    cyclotomic_expand_eq_cyclotomic hp.out hn, map_cyclotomic, mul_comm]
-#align polynomial.cyclotomic_mul_prime_dvd_eq_pow Polynomial.cyclotomic_mul_prime_dvd_eq_pow
-
-/-- If `R` is of characteristic `p` and `¬p ∣ m`, then
-`cyclotomic (p ^ k * m) R = (cyclotomic m R) ^ (p ^ k - p ^ (k - 1))`. -/
-theorem cyclotomic_mul_prime_pow_eq (R : Type _) {p m : ℕ} [Fact (Nat.Prime p)] [Ring R] [CharP R p]
-    (hm : ¬p ∣ m) : ∀ {k}, 0 < k → cyclotomic (p ^ k * m) R = cyclotomic m R ^ (p ^ k - p ^ (k - 1))
-  | 1, _ => by
-    rw [pow_one, Nat.sub_self, pow_zero, mul_comm, cyclotomic_mul_prime_eq_pow_of_not_dvd R hm]
-  | a + 2, _ =>
-    by
-    have hdiv : p ∣ p ^ a.succ * m := ⟨p ^ a * m, by rw [← mul_assoc, pow_succ]⟩
-    rw [pow_succ, mul_assoc, mul_comm, cyclotomic_mul_prime_dvd_eq_pow R hdiv,
-      cyclotomic_mul_prime_pow_eq a.succ_pos, ← pow_mul]
-    congr 1
-    simp only [tsub_zero, Nat.succ_sub_succ_eq_sub]
-    rw [Nat.mul_sub_right_distrib, mul_comm, pow_succ']
-#align polynomial.cyclotomic_mul_prime_pow_eq Polynomial.cyclotomic_mul_prime_pow_eq
-
-/-- If `R` is of characteristic `p` and `¬p ∣ m`, then `ζ` is a root of `cyclotomic (p ^ k * m) R`
- if and only if it is a primitive `m`-th root of unity. -/
-theorem isRoot_cyclotomic_prime_pow_mul_iff_of_charP {m k p : ℕ} {R : Type _} [CommRing R]
-    [IsDomain R] [hp : Fact (Nat.Prime p)] [hchar : CharP R p] {μ : R} [NeZero (m : R)] :
-    (Polynomial.cyclotomic (p ^ k * m) R).IsRoot μ ↔ IsPrimitiveRoot μ m :=
-  by
-  rcases k.eq_zero_or_pos with (rfl | hk)
-  · rw [pow_zero, one_mul, is_root_cyclotomic_iff]
-  refine' ⟨fun h => _, fun h => _⟩
-  · rw [is_root.def, cyclotomic_mul_prime_pow_eq R (NeZero.not_char_dvd R p m) hk, eval_pow] at h
-    replace h := pow_eq_zero h
-    rwa [← is_root.def, is_root_cyclotomic_iff] at h
-  · rw [← is_root_cyclotomic_iff, is_root.def] at h
-    rw [cyclotomic_mul_prime_pow_eq R (NeZero.not_char_dvd R p m) hk, is_root.def, eval_pow, h,
-      zero_pow]
-    simp only [tsub_pos_iff_lt]
-    apply pow_strictMono_right hp.out.one_lt (Nat.pred_lt hk.ne')
-#align polynomial.is_root_cyclotomic_prime_pow_mul_iff_of_char_p Polynomial.isRoot_cyclotomic_prime_pow_mul_iff_of_charP
-
-end CharP
-
 end Polynomial
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -183,7 +183,7 @@ theorem prod_cyclotomic'_eq_x_pow_sub_one {K : Type _} [CommRing K] [IsDomain K]
   by
   have hd : (n.divisors : Set ℕ).PairwiseDisjoint fun k => primitiveRoots k K :=
     fun x hx y hy hne => IsPrimitiveRoot.disjoint hne
-  simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← Finset.prod_bunionᵢ hd,
+  simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← Finset.prod_biUnion hd,
     h.nth_roots_one_eq_bUnion_primitive_roots]
 #align polynomial.prod_cyclotomic'_eq_X_pow_sub_one Polynomial.prod_cyclotomic'_eq_x_pow_sub_one
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/02ba8949f486ebecf93fe7460f1ed0564b5e442c

@@ -810,7 +810,7 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
   apply Nat.coprime.symm
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
-  replace h := (ZMod.nat_coe_zMod_eq_zero_iff_dvd a p).2 h
+  replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h
   rw [is_root.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
   by_cases hone : n = 1
   · simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -498,7 +498,7 @@ theorem isRoot_of_unity_of_root_cyclotomic {n : ℕ} {R} [CommRing R] {ζ : R} {
   have := congr_arg (eval ζ) (prod_cyclotomic_eq_X_pow_sub_one hn R).symm
   rw [eval_sub, eval_pow, eval_X, eval_one] at this
   convert eq_add_of_sub_eq' this
-  convert (add_zero _).symm
+  convert(add_zero _).symm
   apply eval_eq_zero_of_dvd_of_eval_eq_zero _ h
   exact Finset.dvd_prod_of_mem _ hi
 #align polynomial.is_root_of_unity_of_root_cyclotomic Polynomial.isRoot_of_unity_of_root_cyclotomic

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -839,7 +839,7 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
     simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
   rw [is_root.def] at hroot
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
-    Finset.prod_cons, eval_mul, hroot, zero_mul]
+    Finset.prod_cons, eval_mul, hroot, MulZeroClass.zero_mul]
 #align polynomial.order_of_root_cyclotomic_dvd Polynomial.orderOf_root_cyclotomic_dvd
 
 end Order

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

@@ -105,7 +105,7 @@ theorem cyclotomic'_two (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [CharP
 /-- `cyclotomic' n R` is monic. -/
 theorem cyclotomic'.monic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
     (cyclotomic' n R).Monic :=
-  monic_prod_of_monic _ _ fun z hz => monic_x_sub_c _
+  monic_prod_of_monic _ _ fun z hz => monic_X_sub_C _
 #align polynomial.cyclotomic'.monic Polynomial.cyclotomic'.monic
 
 /-- `cyclotomic' n R` is different from `0`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -397,14 +397,14 @@ theorem prod_cyclotomic_eq_x_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [
       Polynomial.map_one, Polynomial.map_sub]
     exact prod_cyclotomic'_eq_X_pow_sub_one hpos (Complex.isPrimitiveRoot_exp n hpos.ne')
   simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one,
-    Polynomial.map_pow, Polynomial.map_x] using congr_arg (map (Int.castRingHom R)) integer
+    Polynomial.map_pow, Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) integer
 #align polynomial.prod_cyclotomic_eq_X_pow_sub_one Polynomial.prod_cyclotomic_eq_x_pow_sub_one
 
 theorem cyclotomic.dvd_x_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomic n R ∣ X ^ n - 1 :=
   by
   suffices cyclotomic n ℤ ∣ X ^ n - 1 by
     simpa only [map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
-      Polynomial.map_x] using map_dvd (Int.castRingHom R) this
+      Polynomial.map_X] using map_dvd (Int.castRingHom R) this
   rcases n.eq_zero_or_pos with (rfl | hn)
   · simp
   rw [← prod_cyclotomic_eq_X_pow_sub_one hn]
@@ -416,7 +416,7 @@ theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
   by
   suffices (∏ i in n.divisors.eraseₓ 1, cyclotomic i ℤ) = ∑ i in Finset.range n, X ^ i by
     simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
-      Polynomial.map_x] using congr_arg (map (Int.castRingHom R)) this
+      Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this
   rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (Nat.one_mem_divisors.2 h.ne'),
     cyclotomic_one, geom_sum_mul, prod_cyclotomic_eq_X_pow_sub_one h]
 #align polynomial.prod_cyclotomic_eq_geom_sum Polynomial.prod_cyclotomic_eq_geom_sum
@@ -426,7 +426,7 @@ theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
     cyclotomic p R = ∑ i in Finset.range p, X ^ i :=
   by
   suffices cyclotomic p ℤ = ∑ i in range p, X ^ i by
-    simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_x] using
+    simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_X] using
       congr_arg (map (Int.castRingHom R)) this
   rw [← prod_cyclotomic_eq_geom_sum hp.out.pos, hp.out.divisors,
     erase_insert (mem_singleton.not.2 hp.out.ne_one.symm), prod_singleton]
@@ -449,7 +449,7 @@ theorem cyclotomic_dvd_geom_sum_of_dvd (R) [Ring R] {d n : ℕ} (hdn : d ∣ n)
     cyclotomic d R ∣ ∑ i in Finset.range n, X ^ i :=
   by
   suffices cyclotomic d ℤ ∣ ∑ i in Finset.range n, X ^ i by
-    simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_x] using
+    simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_X] using
       map_dvd (Int.castRingHom R) this
   rcases n.eq_zero_or_pos with (rfl | hn)
   · simp

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -74,7 +74,7 @@ variable {R : Type _} [CommRing R] [IsDomain R]
 /-- The modified `n`-th cyclotomic polynomial with coefficients in `R`, it is the usual cyclotomic
 polynomial if there is a primitive `n`-th root of unity in `R`. -/
 def cyclotomic' (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] : R[X] :=
-  ∏ μ in primitiveRoots n R, x - c μ
+  ∏ μ in primitiveRoots n R, X - C μ
 #align polynomial.cyclotomic' Polynomial.cyclotomic'
 
 /-- The zeroth modified cyclotomic polyomial is `1`. -/
@@ -85,7 +85,7 @@ theorem cyclotomic'_zero (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 0
 
 /-- The first modified cyclotomic polyomial is `X - 1`. -/
 @[simp]
-theorem cyclotomic'_one (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 1 R = x - 1 := by
+theorem cyclotomic'_one (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 1 R = X - 1 := by
   simp only [cyclotomic', Finset.prod_singleton, RingHom.map_one,
     IsPrimitiveRoot.primitiveRoots_one]
 #align polynomial.cyclotomic'_one Polynomial.cyclotomic'_one
@@ -93,7 +93,7 @@ theorem cyclotomic'_one (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 1 R
 /-- The second modified cyclotomic polyomial is `X + 1` if the characteristic of `R` is not `2`. -/
 @[simp]
 theorem cyclotomic'_two (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [CharP R p] (hp : p ≠ 2) :
-    cyclotomic' 2 R = x + 1 := by
+    cyclotomic' 2 R = X + 1 := by
   rw [cyclotomic']
   have prim_root_two : primitiveRoots 2 R = {(-1 : R)} :=
     by
@@ -143,7 +143,7 @@ theorem roots_of_cyclotomic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
 /-- If there is a primitive `n`th root of unity in `K`, then `X ^ n - 1 = ∏ (X - μ)`, where `μ`
 varies over the `n`-th roots of unity. -/
 theorem x_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
-    x ^ n - 1 = ∏ ζ in nthRootsFinset n R, x - c ζ :=
+    X ^ n - 1 = ∏ ζ in nthRootsFinset n R, X - C ζ :=
   by
   rw [nth_roots_finset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]
   simp only [Finset.prod_mk, RingHom.map_one]
@@ -171,7 +171,7 @@ theorem cyclotomic'_splits (n : ℕ) : Splits (RingHom.id K) (cyclotomic' n K) :
 
 /-- If there is a primitive `n`-th root of unity in `K`, then `X ^ n - 1`splits. -/
 theorem x_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
-    Splits (RingHom.id K) (x ^ n - c (1 : K)) := by
+    Splits (RingHom.id K) (X ^ n - C (1 : K)) := by
   rw [splits_iff_card_roots, ← nth_roots, IsPrimitiveRoot.card_nthRoots h, nat_degree_X_pow_sub_C]
 #align polynomial.X_pow_sub_one_splits Polynomial.x_pow_sub_one_splits
 
@@ -179,7 +179,7 @@ theorem x_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
 `∏ i in nat.divisors n, cyclotomic' i K = X ^ n - 1`. -/
 theorem prod_cyclotomic'_eq_x_pow_sub_one {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
-    (∏ i in Nat.divisors n, cyclotomic' i K) = x ^ n - 1 :=
+    (∏ i in Nat.divisors n, cyclotomic' i K) = X ^ n - 1 :=
   by
   have hd : (n.divisors : Set ℕ).PairwiseDisjoint fun k => primitiveRoots k K :=
     fun x hx y hy hne => IsPrimitiveRoot.disjoint hne
@@ -191,7 +191,7 @@ theorem prod_cyclotomic'_eq_x_pow_sub_one {K : Type _} [CommRing K] [IsDomain K]
 `cyclotomic' n K = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic' i K)`. -/
 theorem cyclotomic'_eq_x_pow_sub_one_div {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
-    cyclotomic' n K = (x ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic' i K :=
+    cyclotomic' n K = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic' i K :=
   by
   rw [← prod_cyclotomic'_eq_X_pow_sub_one hpos h, ← Nat.cons_self_properDivisors hpos.ne',
     Finset.prod_cons]
@@ -331,7 +331,7 @@ theorem cyclotomic_zero (R : Type _) [Ring R] : cyclotomic 0 R = 1 := by
 
 /-- The first cyclotomic polyomial is `X - 1`. -/
 @[simp]
-theorem cyclotomic_one (R : Type _) [Ring R] : cyclotomic 1 R = x - 1 :=
+theorem cyclotomic_one (R : Type _) [Ring R] : cyclotomic 1 R = X - 1 :=
   by
   have hspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ := by
     simp only [cyclotomic'_one, PNat.one_coe, map_X, Polynomial.map_one, Polynomial.map_sub]
@@ -388,7 +388,7 @@ open Finset
 
 /-- `∏ i in nat.divisors n, cyclotomic i R = X ^ n - 1`. -/
 theorem prod_cyclotomic_eq_x_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [CommRing R] :
-    (∏ i in Nat.divisors n, cyclotomic i R) = x ^ n - 1 :=
+    (∏ i in Nat.divisors n, cyclotomic i R) = X ^ n - 1 :=
   by
   have integer : (∏ i in Nat.divisors n, cyclotomic i ℤ) = X ^ n - 1 :=
     by
@@ -400,7 +400,7 @@ theorem prod_cyclotomic_eq_x_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [
     Polynomial.map_pow, Polynomial.map_x] using congr_arg (map (Int.castRingHom R)) integer
 #align polynomial.prod_cyclotomic_eq_X_pow_sub_one Polynomial.prod_cyclotomic_eq_x_pow_sub_one
 
-theorem cyclotomic.dvd_x_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomic n R ∣ x ^ n - 1 :=
+theorem cyclotomic.dvd_x_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomic n R ∣ X ^ n - 1 :=
   by
   suffices cyclotomic n ℤ ∣ X ^ n - 1 by
     simpa only [map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
@@ -412,9 +412,9 @@ theorem cyclotomic.dvd_x_pow_sub_one (n : ℕ) (R : Type _) [Ring R] : cyclotomi
 #align polynomial.cyclotomic.dvd_X_pow_sub_one Polynomial.cyclotomic.dvd_x_pow_sub_one
 
 theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
-    (∏ i in n.divisors.eraseₓ 1, cyclotomic i R) = ∑ i in Finset.range n, x ^ i :=
+    (∏ i in n.divisors.eraseₓ 1, cyclotomic i R) = ∑ i in Finset.range n, X ^ i :=
   by
-  suffices (∏ i in n.divisors.eraseₓ 1, cyclotomic i ℤ) = ∑ i in Finset.range n, x ^ i by
+  suffices (∏ i in n.divisors.eraseₓ 1, cyclotomic i ℤ) = ∑ i in Finset.range n, X ^ i by
     simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
       Polynomial.map_x] using congr_arg (map (Int.castRingHom R)) this
   rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (Nat.one_mem_divisors.2 h.ne'),
@@ -423,7 +423,7 @@ theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
 
 /-- If `p` is prime, then `cyclotomic p R = ∑ i in range p, X ^ i`. -/
 theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
-    cyclotomic p R = ∑ i in Finset.range p, x ^ i :=
+    cyclotomic p R = ∑ i in Finset.range p, X ^ i :=
   by
   suffices cyclotomic p ℤ = ∑ i in range p, X ^ i by
     simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_x] using
@@ -433,20 +433,20 @@ theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
 #align polynomial.cyclotomic_prime Polynomial.cyclotomic_prime
 
 theorem cyclotomic_prime_mul_x_sub_one (R : Type _) [Ring R] (p : ℕ) [hn : Fact (Nat.Prime p)] :
-    cyclotomic p R * (x - 1) = x ^ p - 1 := by rw [cyclotomic_prime, geom_sum_mul]
+    cyclotomic p R * (X - 1) = X ^ p - 1 := by rw [cyclotomic_prime, geom_sum_mul]
 #align polynomial.cyclotomic_prime_mul_X_sub_one Polynomial.cyclotomic_prime_mul_x_sub_one
 
 @[simp]
-theorem cyclotomic_two (R : Type _) [Ring R] : cyclotomic 2 R = x + 1 := by simp [cyclotomic_prime]
+theorem cyclotomic_two (R : Type _) [Ring R] : cyclotomic 2 R = X + 1 := by simp [cyclotomic_prime]
 #align polynomial.cyclotomic_two Polynomial.cyclotomic_two
 
 @[simp]
-theorem cyclotomic_three (R : Type _) [Ring R] : cyclotomic 3 R = x ^ 2 + x + 1 := by
+theorem cyclotomic_three (R : Type _) [Ring R] : cyclotomic 3 R = X ^ 2 + X + 1 := by
   simp [cyclotomic_prime, sum_range_succ']
 #align polynomial.cyclotomic_three Polynomial.cyclotomic_three
 
 theorem cyclotomic_dvd_geom_sum_of_dvd (R) [Ring R] {d n : ℕ} (hdn : d ∣ n) (hd : d ≠ 1) :
-    cyclotomic d R ∣ ∑ i in Finset.range n, x ^ i :=
+    cyclotomic d R ∣ ∑ i in Finset.range n, X ^ i :=
   by
   suffices cyclotomic d ℤ ∣ ∑ i in Finset.range n, X ^ i by
     simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_x] using
@@ -460,7 +460,7 @@ theorem cyclotomic_dvd_geom_sum_of_dvd (R) [Ring R] {d n : ℕ} (hdn : d ∣ n)
 
 theorem x_pow_sub_one_mul_prod_cyclotomic_eq_x_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
     (h : d ∈ n.properDivisors) :
-    ((x ^ d - 1) * ∏ x in n.divisors \ d.divisors, cyclotomic x R) = x ^ n - 1 :=
+    ((X ^ d - 1) * ∏ x in n.divisors \ d.divisors, cyclotomic x R) = X ^ n - 1 :=
   by
   obtain ⟨hd, hdn⟩ := nat.mem_proper_divisors.mp h
   have h0n : 0 < n := pos_of_gt hdn
@@ -470,7 +470,7 @@ theorem x_pow_sub_one_mul_prod_cyclotomic_eq_x_pow_sub_one_of_dvd (R) [CommRing
 #align polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd Polynomial.x_pow_sub_one_mul_prod_cyclotomic_eq_x_pow_sub_one_of_dvd
 
 theorem x_pow_sub_one_mul_cyclotomic_dvd_x_pow_sub_one_of_dvd (R) [CommRing R] {d n : ℕ}
-    (h : d ∈ n.properDivisors) : (x ^ d - 1) * cyclotomic n R ∣ x ^ n - 1 :=
+    (h : d ∈ n.properDivisors) : (X ^ d - 1) * cyclotomic n R ∣ X ^ n - 1 :=
   by
   have hdn := (nat.mem_proper_divisors.mp h).2
   use ∏ x in n.proper_divisors \ d.divisors, cyclotomic x R
@@ -514,7 +514,7 @@ open ArithmeticFunction
 theorem cyclotomic_eq_prod_x_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [CommRing R]
     [IsDomain R] :
     algebraMap _ (Ratfunc R) (cyclotomic n R) =
-      ∏ i in n.divisorsAntidiagonal, algebraMap R[X] _ (x ^ i.snd - 1) ^ μ i.fst :=
+      ∏ i in n.divisorsAntidiagonal, algebraMap R[X] _ (X ^ i.snd - 1) ^ μ i.fst :=
   by
   rcases n.eq_zero_or_pos with (rfl | hpos)
   · simp
@@ -538,7 +538,7 @@ end ArithmeticFunction
 /-- We have
 `cyclotomic n R = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic i K)`. -/
 theorem cyclotomic_eq_x_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) :
-    cyclotomic n R = (x ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R :=
+    cyclotomic n R = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R :=
   by
   nontriviality R
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos, ← Nat.cons_self_properDivisors hpos.ne',
@@ -560,7 +560,7 @@ theorem cyclotomic_eq_x_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpo
 /-- If `m` is a proper divisor of `n`, then `X ^ m - 1` divides
 `∏ i in nat.proper_divisors n, cyclotomic i R`. -/
 theorem x_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ} (hpos : 0 < n)
-    (hm : m ∣ n) (hdiff : m ≠ n) : x ^ m - 1 ∣ ∏ i in Nat.properDivisors n, cyclotomic i R :=
+    (hm : m ∣ n) (hdiff : m ≠ n) : X ^ m - 1 ∣ ∏ i in Nat.properDivisors n, cyclotomic i R :=
   by
   replace hm :=
     Nat.mem_properDivisors.2
@@ -578,7 +578,7 @@ theorem x_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ}
 `cyclotomic n K = ∏ μ in primitive_roots n R, (X - C μ)`. In particular,
 `cyclotomic n K = cyclotomic' n K` -/
 theorem cyclotomic_eq_prod_x_sub_primitiveRoots {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
-    {n : ℕ} (hz : IsPrimitiveRoot ζ n) : cyclotomic n K = ∏ μ in primitiveRoots n K, x - c μ :=
+    {n : ℕ} (hz : IsPrimitiveRoot ζ n) : cyclotomic n K = ∏ μ in primitiveRoots n K, X - C μ :=
   by
   rw [← cyclotomic']
   induction' n using Nat.strong_induction_on with k hk generalizing ζ hz
@@ -713,7 +713,7 @@ theorem cyclotomic_injective {R : Type _} [CommRing R] [CharZero R] :
 #align polynomial.cyclotomic_injective Polynomial.cyclotomic_injective
 
 theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P : R[X]) :
-    P = cyclotomic n R ↔ (P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R) = x ^ n - 1 :=
+    P = cyclotomic n R ↔ (P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R) = X ^ n - 1 :=
   by
   nontriviality R
   refine' ⟨fun hcycl => _, fun hP => _⟩
@@ -735,7 +735,7 @@ theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P
 /-- If `p ^ k` is a prime power, then
 `cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ (p ^ n)) ^ i`. -/
 theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (hp : p.Prime) :
-    cyclotomic (p ^ (n + 1)) R = ∑ i in Finset.range p, (x ^ p ^ n) ^ i :=
+    cyclotomic (p ^ (n + 1)) R = ∑ i in Finset.range p, (X ^ p ^ n) ^ i :=
   by
   have :
     ∀ m,
@@ -759,7 +759,7 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
 
 theorem cyclotomic_prime_pow_mul_x_pow_sub_one (R : Type _) [CommRing R] (p k : ℕ)
     [hn : Fact (Nat.Prime p)] :
-    cyclotomic (p ^ (k + 1)) R * (x ^ p ^ k - 1) = x ^ p ^ (k + 1) - 1 := by
+    cyclotomic (p ^ (k + 1)) R * (X ^ p ^ k - 1) = X ^ p ^ (k + 1) - 1 := by
   rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, ← pow_mul, pow_succ, mul_comm]
 #align polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one Polynomial.cyclotomic_prime_pow_mul_x_pow_sub_one
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

This is a PR that's a temporary measure to improve performance of congr!/convert, and the implementation may change in a future PR with a new version of congr!.

Introduces two typeclasses that are meant to quickly evaluate in common cases of Subsingleton and IsEmpty. Makes congr! use these typeclasses rather than Subsingleton.

Local Subsingleton/IsEmpty instances are included as Fast instances. To get congr!/convert to reason about subsingleton types, you can add such instances to the local context. Or, you can apply Subsingleton.elim yourself.

Zulip discussion

@@ -291,6 +291,7 @@ theorem int_cyclotomic_unique {n : ℕ} {P : ℤ[X]} (h : map (Int.castRingHom 
 theorem map_cyclotomic (n : ℕ) {R S : Type*} [Ring R] [Ring S] (f : R →+* S) :
     map f (cyclotomic n R) = cyclotomic n S := by
   rw [← map_cyclotomic_int n R, ← map_cyclotomic_int n S, map_map]
+  have : Subsingleton (ℤ →+* S) := inferInstance
   congr!
 #align polynomial.map_cyclotomic Polynomial.map_cyclotomic
 

Now that I am defining NNRat.cast, I want a definitive answer to this naming issue. Plenty of lemmas in mathlib already use natCast/intCast/ratCast over nat_cast/int_cast/rat_cast, and this matches with the general expectation that underscore-separated name parts correspond to a single declaration.

@@ -618,7 +618,7 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
   apply Nat.Coprime.symm
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
-  replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h
+  replace h := (ZMod.natCast_zmod_eq_zero_iff_dvd a p).2 h
   rw [IsRoot.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
   by_cases hone : n = 1
   · simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -619,7 +619,7 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h
-  rw [IsRoot.definition, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
+  rw [IsRoot.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
   by_cases hone : n = 1
   · simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
       one_ne_zero, coeff_sub] at hroot
@@ -642,7 +642,7 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
     simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
     apply Units.val_eq_one.1
     simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
-  rw [IsRoot.definition] at hroot
+  rw [IsRoot.def] at hroot
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
     Finset.prod_cons, eval_mul, hroot, zero_mul]
 #align polynomial.order_of_root_cyclotomic_dvd Polynomial.orderOf_root_cyclotomic_dvd

Polynomial and MvPolynomial are algebraic objects, hence should be under Algebra (or at least not under Data)

@@ -5,9 +5,9 @@ Authors: Riccardo Brasca
 -/
 import Mathlib.Algebra.NeZero
 import Mathlib.Algebra.Polynomial.BigOperators
+import Mathlib.Algebra.Polynomial.Lifts
+import Mathlib.Algebra.Polynomial.Splits
 import Mathlib.RingTheory.RootsOfUnity.Complex
-import Mathlib.Data.Polynomial.Lifts
-import Mathlib.Data.Polynomial.Splits
 import Mathlib.FieldTheory.RatFunc
 import Mathlib.NumberTheory.ArithmeticFunction
 import Mathlib.RingTheory.RootsOfUnity.Basic

@@ -224,7 +224,7 @@ theorem int_coeff_of_cyclotomic' {K : Type*} [CommRing K] [IsDomain K] {ζ : K}
     constructor
     · rw [zero_add, mul_comm, ← prod_cyclotomic'_eq_X_pow_sub_one hpos h, ←
         Nat.cons_self_properDivisors hpos.ne', Finset.prod_cons]
-    · simpa only [degree_zero, bot_lt_iff_ne_bot, Ne.def, degree_eq_bot] using Bmo.ne_zero
+    · simpa only [degree_zero, bot_lt_iff_ne_bot, Ne, degree_eq_bot] using Bmo.ne_zero
   replace huniq := div_modByMonic_unique (cyclotomic' k K) (0 : K[X]) Bmo huniq
   simp only [lifts, RingHom.mem_rangeS]
   use Q₁
@@ -341,7 +341,7 @@ theorem degree_cyclotomic (n : ℕ) (R : Type*) [Ring R] [Nontrivial R] :
     rw [← degree_cyclotomic' (Complex.isPrimitiveRoot_exp k.succ (Nat.succ_ne_zero k))]
     exact (int_cyclotomic_spec k.succ).2.1
   simp only [(int_cyclotomic_spec n).right.right, eq_intCast, Monic.leadingCoeff, Int.cast_one,
-    Ne.def, not_false_iff, one_ne_zero]
+    Ne, not_false_iff, one_ne_zero]
 #align polynomial.degree_cyclotomic Polynomial.degree_cyclotomic
 
 /-- The natural degree of `cyclotomic n` is `totient n`. -/
@@ -471,7 +471,7 @@ theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type*) [Comm
     intro n hn
     rw [← prod_cyclotomic_eq_X_pow_sub_one hn R, map_prod]
   rw [(prod_eq_iff_prod_pow_moebius_eq_of_nonzero (fun n hn => _) fun n hn => _).1 h n hpos] <;>
-    simp_rw [Ne.def, IsFractionRing.to_map_eq_zero_iff]
+    simp_rw [Ne, IsFractionRing.to_map_eq_zero_iff]
   · simp [cyclotomic_ne_zero]
   · intro n hn
     apply Monic.ne_zero

This will become an error in 2024-03-16 nightly, possibly not permanently.

Co-authored-by: Scott Morrison <scott@tqft.net>

@@ -619,7 +619,7 @@ theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime :
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h
-  rw [IsRoot.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
+  rw [IsRoot.definition, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
   by_cases hone : n = 1
   · simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
       one_ne_zero, coeff_sub] at hroot
@@ -642,7 +642,7 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
     simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
     apply Units.val_eq_one.1
     simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
-  rw [IsRoot.def] at hroot
+  rw [IsRoot.definition] at hroot
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
     Finset.prod_cons, eval_mul, hroot, zero_mul]
 #align polynomial.order_of_root_cyclotomic_dvd Polynomial.orderOf_root_cyclotomic_dvd

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -638,8 +638,8 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
     (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) :
     orderOf (ZMod.unitOfCoprime a (coprime_of_root_cyclotomic hpos hroot)) ∣ n := by
   apply orderOf_dvd_of_pow_eq_one
-  suffices hpow : eval (Nat.castRingHom (ZMod p) a) (X ^ n - 1 : (ZMod p)[X]) = 0
-  · simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
+  suffices hpow : eval (Nat.castRingHom (ZMod p) a) (X ^ n - 1 : (ZMod p)[X]) = 0 by
+    simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
     apply Units.val_eq_one.1
     simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
   rw [IsRoot.def] at hroot

This changes Nat.ArithmeticFunction to ArithmeticFunction since the Nat part seems redundant.

See here on Zulip.

@@ -455,9 +455,9 @@ set_option linter.uppercaseLean3 false in
 
 section ArithmeticFunction
 
-open Nat.ArithmeticFunction
+open ArithmeticFunction
 
-open scoped Nat.ArithmeticFunction
+open scoped ArithmeticFunction
 
 /-- `cyclotomic n R` can be expressed as a product in a fraction field of `R[X]`
   using Möbius inversion. -/

This involves moving lemmas from Algebra.GroupPower.Ring to Algebra.GroupWithZero.Basic and changing some 0 < n assumptions to n ≠ 0.

From LeanAPAP

@@ -606,8 +606,7 @@ theorem cyclotomic_coeff_zero (R : Type*) [CommRing R] {n : ℕ} (hn : 1 < n) :
       hprod, mul_neg, mul_one]
   have hzero : (X ^ n - 1 : R[X]).coeff 0 = (-1 : R) := by
     rw [coeff_zero_eq_eval_zero _]
-    simp only [zero_pow (lt_of_lt_of_le zero_lt_two hn), eval_X, eval_one, zero_sub, eval_pow,
-      eval_sub]
+    simp only [zero_pow (by positivity : n ≠ 0), eval_X, eval_one, zero_sub, eval_pow, eval_sub]
   rw [hzero] at heq
   exact neg_inj.mp (Eq.symm heq)
 #align polynomial.cyclotomic_coeff_zero Polynomial.cyclotomic_coeff_zero

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -8,9 +8,7 @@ import Mathlib.Algebra.Polynomial.BigOperators
 import Mathlib.RingTheory.RootsOfUnity.Complex
 import Mathlib.Data.Polynomial.Lifts
 import Mathlib.Data.Polynomial.Splits
-import Mathlib.Data.ZMod.Algebra
 import Mathlib.FieldTheory.RatFunc
-import Mathlib.FieldTheory.Separable
 import Mathlib.NumberTheory.ArithmeticFunction
 import Mathlib.RingTheory.RootsOfUnity.Basic
 

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -133,14 +133,14 @@ varies over the `n`-th roots of unity. -/
 theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     X ^ n - 1 = ∏ ζ in nthRootsFinset n R, (X - C ζ) := by
   classical
-  rw [nthRootsFinset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]
+  rw [nthRootsFinset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_one_nodup h)]
   simp only [Finset.prod_mk, RingHom.map_one]
   rw [nthRoots]
   have hmonic : (X ^ n - C (1 : R)).Monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm
   symm
   apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmonic
   rw [@natDegree_X_pow_sub_C R _ _ n 1, ← nthRoots]
-  exact IsPrimitiveRoot.card_nthRoots h
+  exact IsPrimitiveRoot.card_nthRoots_one h
 set_option linter.uppercaseLean3 false in
 #align polynomial.X_pow_sub_one_eq_prod Polynomial.X_pow_sub_one_eq_prod
 
@@ -160,7 +160,7 @@ theorem cyclotomic'_splits (n : ℕ) : Splits (RingHom.id K) (cyclotomic' n K) :
 /-- If there is a primitive `n`-th root of unity in `K`, then `X ^ n - 1` splits. -/
 theorem X_pow_sub_one_splits {ζ : K} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
     Splits (RingHom.id K) (X ^ n - C (1 : K)) := by
-  rw [splits_iff_card_roots, ← nthRoots, IsPrimitiveRoot.card_nthRoots h, natDegree_X_pow_sub_C]
+  rw [splits_iff_card_roots, ← nthRoots, IsPrimitiveRoot.card_nthRoots_one h, natDegree_X_pow_sub_C]
 set_option linter.uppercaseLean3 false in
 #align polynomial.X_pow_sub_one_splits Polynomial.X_pow_sub_one_splits
 

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -339,7 +339,7 @@ theorem degree_cyclotomic (n : ℕ) (R : Type*) [Ring R] [Nontrivial R] :
   rw [← map_cyclotomic_int]
   rw [degree_map_eq_of_leadingCoeff_ne_zero (Int.castRingHom R) _]
   · cases' n with k
-    · simp only [cyclotomic, degree_one, dif_pos, Nat.totient_zero, WithTop.coe_zero]
+    · simp only [cyclotomic, degree_one, dif_pos, Nat.totient_zero, CharP.cast_eq_zero]
     rw [← degree_cyclotomic' (Complex.isPrimitiveRoot_exp k.succ (Nat.succ_ne_zero k))]
     exact (int_cyclotomic_spec k.succ).2.1
   simp only [(int_cyclotomic_spec n).right.right, eq_intCast, Monic.leadingCoeff, Int.cast_one,

This doesn't change any polynomial operations, but:

• Makes some Decidable values computable (otherwise, they're pointless!)
• Add a few missing arguments to lemmas here and there to make them more general

This is exhaustive, within the directories it touches.

Once again, the use of letI := Classical.decEq R instead of classical here is because of the weird style of proofs in these files, where if is preferred to by_cases.

@@ -49,7 +49,7 @@ to get a polynomial with integer coefficients and then we map it to `R[X]`, for
 -/
 
 
-open scoped Classical BigOperators Polynomial
+open scoped BigOperators Polynomial
 
 noncomputable section
 
@@ -132,6 +132,7 @@ theorem roots_of_cyclotomic (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] :
 varies over the `n`-th roots of unity. -/
 theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     X ^ n - 1 = ∏ ζ in nthRootsFinset n R, (X - C ζ) := by
+  classical
   rw [nthRootsFinset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]
   simp only [Finset.prod_mk, RingHom.map_one]
   rw [nthRoots]
@@ -168,6 +169,7 @@ set_option linter.uppercaseLean3 false in
 theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type*} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     ∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1 := by
+  classical
   have hd : (n.divisors : Set ℕ).PairwiseDisjoint fun k => primitiveRoots k K :=
     fun x _ y _ hne => IsPrimitiveRoot.disjoint hne
   simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← Finset.prod_biUnion hd,

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -615,8 +615,8 @@ theorem cyclotomic_coeff_zero (R : Type*) [CommRing R] {n : ℕ} (hn : 1 < n) :
 /-- If `(a : ℕ)` is a root of `cyclotomic n (ZMod p)`, where `p` is a prime, then `a` and `p` are
 coprime. -/
 theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime : Fact p.Prime] {a : ℕ}
-    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.coprime p := by
-  apply Nat.coprime.symm
+    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.Coprime p := by
+  apply Nat.Coprime.symm
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h

This reverts commit 6f8e8104. Unfortunately this bump was not linted correctly, as CI did not run runLinter Mathlib.

We can unrevert once that's fixed.

@@ -615,8 +615,8 @@ theorem cyclotomic_coeff_zero (R : Type*) [CommRing R] {n : ℕ} (hn : 1 < n) :
 /-- If `(a : ℕ)` is a root of `cyclotomic n (ZMod p)`, where `p` is a prime, then `a` and `p` are
 coprime. -/
 theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime : Fact p.Prime] {a : ℕ}
-    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.Coprime p := by
-  apply Nat.Coprime.symm
+    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.coprime p := by
+  apply Nat.coprime.symm
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h

Some changes have already been review and delegated in #6910 and #7148.

The diff that needs looking at is https://github.com/leanprover-community/mathlib4/pull/7174/commits/64d6d07ee18163627c8f517eb31455411921c5ac

The std bump PR was insta-merged already!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -615,8 +615,8 @@ theorem cyclotomic_coeff_zero (R : Type*) [CommRing R] {n : ℕ} (hn : 1 < n) :
 /-- If `(a : ℕ)` is a root of `cyclotomic n (ZMod p)`, where `p` is a prime, then `a` and `p` are
 coprime. -/
 theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime : Fact p.Prime] {a : ℕ}
-    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.coprime p := by
-  apply Nat.coprime.symm
+    (hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.Coprime p := by
+  apply Nat.Coprime.symm
   rw [hprime.1.coprime_iff_not_dvd]
   intro h
   replace h := (ZMod.nat_cast_zmod_eq_zero_iff_dvd a p).2 h

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -645,7 +645,7 @@ theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.P
     simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
   rw [IsRoot.def] at hroot
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
-    Finset.prod_cons, eval_mul, hroot, MulZeroClass.zero_mul]
+    Finset.prod_cons, eval_mul, hroot, zero_mul]
 #align polynomial.order_of_root_cyclotomic_dvd Polynomial.orderOf_root_cyclotomic_dvd
 
 end Order

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -61,30 +61,30 @@ section Cyclotomic'
 
 section IsDomain
 
-variable {R : Type _} [CommRing R] [IsDomain R]
+variable {R : Type*} [CommRing R] [IsDomain R]
 
 /-- The modified `n`-th cyclotomic polynomial with coefficients in `R`, it is the usual cyclotomic
 polynomial if there is a primitive `n`-th root of unity in `R`. -/
-def cyclotomic' (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] : R[X] :=
+def cyclotomic' (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] : R[X] :=
   ∏ μ in primitiveRoots n R, (X - C μ)
 #align polynomial.cyclotomic' Polynomial.cyclotomic'
 
 /-- The zeroth modified cyclotomic polyomial is `1`. -/
 @[simp]
-theorem cyclotomic'_zero (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 0 R = 1 := by
+theorem cyclotomic'_zero (R : Type*) [CommRing R] [IsDomain R] : cyclotomic' 0 R = 1 := by
   simp only [cyclotomic', Finset.prod_empty, primitiveRoots_zero]
 #align polynomial.cyclotomic'_zero Polynomial.cyclotomic'_zero
 
 /-- The first modified cyclotomic polyomial is `X - 1`. -/
 @[simp]
-theorem cyclotomic'_one (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' 1 R = X - 1 := by
+theorem cyclotomic'_one (R : Type*) [CommRing R] [IsDomain R] : cyclotomic' 1 R = X - 1 := by
   simp only [cyclotomic', Finset.prod_singleton, RingHom.map_one,
     IsPrimitiveRoot.primitiveRoots_one]
 #align polynomial.cyclotomic'_one Polynomial.cyclotomic'_one
 
 /-- The second modified cyclotomic polyomial is `X + 1` if the characteristic of `R` is not `2`. -/
 @[simp]
-theorem cyclotomic'_two (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [CharP R p] (hp : p ≠ 2) :
+theorem cyclotomic'_two (R : Type*) [CommRing R] [IsDomain R] (p : ℕ) [CharP R p] (hp : p ≠ 2) :
     cyclotomic' 2 R = X + 1 := by
   rw [cyclotomic']
   have prim_root_two : primitiveRoots 2 R = {(-1 : R)} := by
@@ -94,13 +94,13 @@ theorem cyclotomic'_two (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [CharP
 #align polynomial.cyclotomic'_two Polynomial.cyclotomic'_two
 
 /-- `cyclotomic' n R` is monic. -/
-theorem cyclotomic'.monic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
+theorem cyclotomic'.monic (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] :
     (cyclotomic' n R).Monic :=
   monic_prod_of_monic _ _ fun _ _ => monic_X_sub_C _
 #align polynomial.cyclotomic'.monic Polynomial.cyclotomic'.monic
 
 /-- `cyclotomic' n R` is different from `0`. -/
-theorem cyclotomic'_ne_zero (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] : cyclotomic' n R ≠ 0 :=
+theorem cyclotomic'_ne_zero (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] : cyclotomic' n R ≠ 0 :=
   (cyclotomic'.monic n R).ne_zero
 #align polynomial.cyclotomic'_ne_zero Polynomial.cyclotomic'_ne_zero
 
@@ -123,7 +123,7 @@ theorem degree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
 #align polynomial.degree_cyclotomic' Polynomial.degree_cyclotomic'
 
 /-- The roots of `cyclotomic' n R` are the primitive `n`-th roots of unity. -/
-theorem roots_of_cyclotomic (n : ℕ) (R : Type _) [CommRing R] [IsDomain R] :
+theorem roots_of_cyclotomic (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] :
     (cyclotomic' n R).roots = (primitiveRoots n R).val := by
   rw [cyclotomic']; exact roots_prod_X_sub_C (primitiveRoots n R)
 #align polynomial.roots_of_cyclotomic Polynomial.roots_of_cyclotomic
@@ -147,7 +147,7 @@ end IsDomain
 
 section Field
 
-variable {K : Type _} [Field K]
+variable {K : Type*} [Field K]
 
 /-- `cyclotomic' n K` splits. -/
 theorem cyclotomic'_splits (n : ℕ) : Splits (RingHom.id K) (cyclotomic' n K) := by
@@ -165,7 +165,7 @@ set_option linter.uppercaseLean3 false in
 
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1`. -/
-theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
+theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type*} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     ∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1 := by
   have hd : (n.divisors : Set ℕ).PairwiseDisjoint fun k => primitiveRoots k K :=
@@ -177,7 +177,7 @@ set_option linter.uppercaseLean3 false in
 
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `cyclotomic' n K = (X ^ k - 1) /ₘ (∏ i in Nat.properDivisors k, cyclotomic' i K)`. -/
-theorem cyclotomic'_eq_X_pow_sub_one_div {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
+theorem cyclotomic'_eq_X_pow_sub_one_div {K : Type*} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
     cyclotomic' n K = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic' i K := by
   rw [← prod_cyclotomic'_eq_X_pow_sub_one hpos h, ← Nat.cons_self_properDivisors hpos.ne',
@@ -197,7 +197,7 @@ set_option linter.uppercaseLean3 false in
 
 /-- If there is a primitive `n`-th root of unity in `K`, then `cyclotomic' n K` comes from a
 monic polynomial with integer coefficients. -/
-theorem int_coeff_of_cyclotomic' {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
+theorem int_coeff_of_cyclotomic' {K : Type*} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (h : IsPrimitiveRoot ζ n) : ∃ P : ℤ[X], map (Int.castRingHom K) P =
       cyclotomic' n K ∧ P.degree = (cyclotomic' n K).degree ∧ P.Monic := by
   refine' lifts_and_degree_eq_and_monic _ (cyclotomic'.monic n K)
@@ -234,7 +234,7 @@ theorem int_coeff_of_cyclotomic' {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
 
 /-- If `K` is of characteristic `0` and there is a primitive `n`-th root of unity in `K`,
 then `cyclotomic n K` comes from a unique polynomial with integer coefficients. -/
-theorem unique_int_coeff_of_cycl {K : Type _} [CommRing K] [IsDomain K] [CharZero K] {ζ : K}
+theorem unique_int_coeff_of_cycl {K : Type*} [CommRing K] [IsDomain K] [CharZero K] {ζ : K}
     {n : ℕ+} (h : IsPrimitiveRoot ζ n) :
     ∃! P : ℤ[X], map (Int.castRingHom K) P = cyclotomic' n K := by
   obtain ⟨P, hP⟩ := int_coeff_of_cyclotomic' h
@@ -250,7 +250,7 @@ end Cyclotomic'
 section Cyclotomic
 
 /-- The `n`-th cyclotomic polynomial with coefficients in `R`. -/
-def cyclotomic (n : ℕ) (R : Type _) [Ring R] : R[X] :=
+def cyclotomic (n : ℕ) (R : Type*) [Ring R] : R[X] :=
   if h : n = 0 then 1
   else map (Int.castRingHom R) (int_coeff_of_cyclotomic' (Complex.isPrimitiveRoot_exp n h)).choose
 #align polynomial.cyclotomic Polynomial.cyclotomic
@@ -263,7 +263,7 @@ theorem int_cyclotomic_rw {n : ℕ} (h : n ≠ 0) :
 #align polynomial.int_cyclotomic_rw Polynomial.int_cyclotomic_rw
 
 /-- `cyclotomic n R` comes from `cyclotomic n ℤ`. -/
-theorem map_cyclotomic_int (n : ℕ) (R : Type _) [Ring R] :
+theorem map_cyclotomic_int (n : ℕ) (R : Type*) [Ring R] :
     map (Int.castRingHom R) (cyclotomic n ℤ) = cyclotomic n R := by
   by_cases hzero : n = 0
   · simp only [hzero, cyclotomic, dif_pos, Polynomial.map_one]
@@ -288,26 +288,26 @@ theorem int_cyclotomic_unique {n : ℕ} {P : ℤ[X]} (h : map (Int.castRingHom 
 
 /-- The definition of `cyclotomic n R` commutes with any ring homomorphism. -/
 @[simp]
-theorem map_cyclotomic (n : ℕ) {R S : Type _} [Ring R] [Ring S] (f : R →+* S) :
+theorem map_cyclotomic (n : ℕ) {R S : Type*} [Ring R] [Ring S] (f : R →+* S) :
     map f (cyclotomic n R) = cyclotomic n S := by
   rw [← map_cyclotomic_int n R, ← map_cyclotomic_int n S, map_map]
   congr!
 #align polynomial.map_cyclotomic Polynomial.map_cyclotomic
 
-theorem cyclotomic.eval_apply {R S : Type _} (q : R) (n : ℕ) [Ring R] [Ring S] (f : R →+* S) :
+theorem cyclotomic.eval_apply {R S : Type*} (q : R) (n : ℕ) [Ring R] [Ring S] (f : R →+* S) :
     eval (f q) (cyclotomic n S) = f (eval q (cyclotomic n R)) := by
   rw [← map_cyclotomic n f, eval_map, eval₂_at_apply]
 #align polynomial.cyclotomic.eval_apply Polynomial.cyclotomic.eval_apply
 
 /-- The zeroth cyclotomic polyomial is `1`. -/
 @[simp]
-theorem cyclotomic_zero (R : Type _) [Ring R] : cyclotomic 0 R = 1 := by
+theorem cyclotomic_zero (R : Type*) [Ring R] : cyclotomic 0 R = 1 := by
   simp only [cyclotomic, dif_pos]
 #align polynomial.cyclotomic_zero Polynomial.cyclotomic_zero
 
 /-- The first cyclotomic polyomial is `X - 1`. -/
 @[simp]
-theorem cyclotomic_one (R : Type _) [Ring R] : cyclotomic 1 R = X - 1 := by
+theorem cyclotomic_one (R : Type*) [Ring R] : cyclotomic 1 R = X - 1 := by
   have hspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ := by
     simp only [cyclotomic'_one, PNat.one_coe, map_X, Polynomial.map_one, Polynomial.map_sub]
   symm
@@ -316,23 +316,23 @@ theorem cyclotomic_one (R : Type _) [Ring R] : cyclotomic 1 R = X - 1 := by
 #align polynomial.cyclotomic_one Polynomial.cyclotomic_one
 
 /-- `cyclotomic n` is monic. -/
-theorem cyclotomic.monic (n : ℕ) (R : Type _) [Ring R] : (cyclotomic n R).Monic := by
+theorem cyclotomic.monic (n : ℕ) (R : Type*) [Ring R] : (cyclotomic n R).Monic := by
   rw [← map_cyclotomic_int]
   exact (int_cyclotomic_spec n).2.2.map _
 #align polynomial.cyclotomic.monic Polynomial.cyclotomic.monic
 
 /-- `cyclotomic n` is primitive. -/
-theorem cyclotomic.isPrimitive (n : ℕ) (R : Type _) [CommRing R] : (cyclotomic n R).IsPrimitive :=
+theorem cyclotomic.isPrimitive (n : ℕ) (R : Type*) [CommRing R] : (cyclotomic n R).IsPrimitive :=
   (cyclotomic.monic n R).isPrimitive
 #align polynomial.cyclotomic.is_primitive Polynomial.cyclotomic.isPrimitive
 
 /-- `cyclotomic n R` is different from `0`. -/
-theorem cyclotomic_ne_zero (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] : cyclotomic n R ≠ 0 :=
+theorem cyclotomic_ne_zero (n : ℕ) (R : Type*) [Ring R] [Nontrivial R] : cyclotomic n R ≠ 0 :=
   (cyclotomic.monic n R).ne_zero
 #align polynomial.cyclotomic_ne_zero Polynomial.cyclotomic_ne_zero
 
 /-- The degree of `cyclotomic n` is `totient n`. -/
-theorem degree_cyclotomic (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] :
+theorem degree_cyclotomic (n : ℕ) (R : Type*) [Ring R] [Nontrivial R] :
     (cyclotomic n R).degree = Nat.totient n := by
   rw [← map_cyclotomic_int]
   rw [degree_map_eq_of_leadingCoeff_ne_zero (Int.castRingHom R) _]
@@ -345,13 +345,13 @@ theorem degree_cyclotomic (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] :
 #align polynomial.degree_cyclotomic Polynomial.degree_cyclotomic
 
 /-- The natural degree of `cyclotomic n` is `totient n`. -/
-theorem natDegree_cyclotomic (n : ℕ) (R : Type _) [Ring R] [Nontrivial R] :
+theorem natDegree_cyclotomic (n : ℕ) (R : Type*) [Ring R] [Nontrivial R] :
     (cyclotomic n R).natDegree = Nat.totient n := by
   rw [natDegree, degree_cyclotomic]; norm_cast
 #align polynomial.nat_degree_cyclotomic Polynomial.natDegree_cyclotomic
 
 /-- The degree of `cyclotomic n R` is positive. -/
-theorem degree_cyclotomic_pos (n : ℕ) (R : Type _) (hpos : 0 < n) [Ring R] [Nontrivial R] :
+theorem degree_cyclotomic_pos (n : ℕ) (R : Type*) (hpos : 0 < n) [Ring R] [Nontrivial R] :
     0 < (cyclotomic n R).degree := by
   rw [degree_cyclotomic n R, Nat.cast_pos]; exact Nat.totient_pos hpos
 #align polynomial.degree_cyclotomic_pos Polynomial.degree_cyclotomic_pos
@@ -359,7 +359,7 @@ theorem degree_cyclotomic_pos (n : ℕ) (R : Type _) (hpos : 0 < n) [Ring R] [No
 open Finset
 
 /-- `∏ i in Nat.divisors n, cyclotomic i R = X ^ n - 1`. -/
-theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [CommRing R] :
+theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type*) [CommRing R] :
     ∏ i in Nat.divisors n, cyclotomic i R = X ^ n - 1 := by
   have integer : ∏ i in Nat.divisors n, cyclotomic i ℤ = X ^ n - 1 := by
     apply map_injective (Int.castRingHom ℂ) Int.cast_injective
@@ -371,7 +371,7 @@ theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [
 set_option linter.uppercaseLean3 false in
 #align polynomial.prod_cyclotomic_eq_X_pow_sub_one Polynomial.prod_cyclotomic_eq_X_pow_sub_one
 
-theorem cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type _) [Ring R] :
+theorem cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type*) [Ring R] :
     cyclotomic n R ∣ X ^ n - 1 := by
   suffices cyclotomic n ℤ ∣ X ^ n - 1 by
     simpa only [map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
@@ -393,7 +393,7 @@ theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
 #align polynomial.prod_cyclotomic_eq_geom_sum Polynomial.prod_cyclotomic_eq_geom_sum
 
 /-- If `p` is prime, then `cyclotomic p R = ∑ i in range p, X ^ i`. -/
-theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
+theorem cyclotomic_prime (R : Type*) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
     cyclotomic p R = ∑ i in Finset.range p, X ^ i := by
   suffices cyclotomic p ℤ = ∑ i in range p, X ^ i by
     simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_X] using
@@ -402,17 +402,17 @@ theorem cyclotomic_prime (R : Type _) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
     erase_insert (mem_singleton.not.2 hp.out.ne_one.symm), prod_singleton]
 #align polynomial.cyclotomic_prime Polynomial.cyclotomic_prime
 
-theorem cyclotomic_prime_mul_X_sub_one (R : Type _) [Ring R] (p : ℕ) [hn : Fact (Nat.Prime p)] :
+theorem cyclotomic_prime_mul_X_sub_one (R : Type*) [Ring R] (p : ℕ) [hn : Fact (Nat.Prime p)] :
     cyclotomic p R * (X - 1) = X ^ p - 1 := by rw [cyclotomic_prime, geom_sum_mul]
 set_option linter.uppercaseLean3 false in
 #align polynomial.cyclotomic_prime_mul_X_sub_one Polynomial.cyclotomic_prime_mul_X_sub_one
 
 @[simp]
-theorem cyclotomic_two (R : Type _) [Ring R] : cyclotomic 2 R = X + 1 := by simp [cyclotomic_prime]
+theorem cyclotomic_two (R : Type*) [Ring R] : cyclotomic 2 R = X + 1 := by simp [cyclotomic_prime]
 #align polynomial.cyclotomic_two Polynomial.cyclotomic_two
 
 @[simp]
-theorem cyclotomic_three (R : Type _) [Ring R] : cyclotomic 3 R = X ^ 2 + X + 1 := by
+theorem cyclotomic_three (R : Type*) [Ring R] : cyclotomic 3 R = X ^ 2 + X + 1 := by
   simp [cyclotomic_prime, sum_range_succ']
 #align polynomial.cyclotomic_three Polynomial.cyclotomic_three
 
@@ -461,7 +461,7 @@ open scoped Nat.ArithmeticFunction
 
 /-- `cyclotomic n R` can be expressed as a product in a fraction field of `R[X]`
   using Möbius inversion. -/
-theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type _) [CommRing R]
+theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type*) [CommRing R]
     [IsDomain R] : algebraMap _ (RatFunc R) (cyclotomic n R) =
       ∏ i in n.divisorsAntidiagonal, algebraMap R[X] _ (X ^ i.snd - 1) ^ μ i.fst := by
   rcases n.eq_zero_or_pos with (rfl | hpos)
@@ -483,7 +483,7 @@ end ArithmeticFunction
 
 /-- We have
 `cyclotomic n R = (X ^ k - 1) /ₘ (∏ i in Nat.properDivisors k, cyclotomic i K)`. -/
-theorem cyclotomic_eq_X_pow_sub_one_div {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) :
+theorem cyclotomic_eq_X_pow_sub_one_div {R : Type*} [CommRing R] {n : ℕ} (hpos : 0 < n) :
     cyclotomic n R = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R := by
   nontriviality R
   rw [← prod_cyclotomic_eq_X_pow_sub_one hpos, ← Nat.cons_self_properDivisors hpos.ne',
@@ -504,7 +504,7 @@ set_option linter.uppercaseLean3 false in
 
 /-- If `m` is a proper divisor of `n`, then `X ^ m - 1` divides
 `∏ i in Nat.properDivisors n, cyclotomic i R`. -/
-theorem X_pow_sub_one_dvd_prod_cyclotomic (R : Type _) [CommRing R] {n m : ℕ} (hpos : 0 < n)
+theorem X_pow_sub_one_dvd_prod_cyclotomic (R : Type*) [CommRing R] {n m : ℕ} (hpos : 0 < n)
     (hm : m ∣ n) (hdiff : m ≠ n) : X ^ m - 1 ∣ ∏ i in Nat.properDivisors n, cyclotomic i R := by
   replace hm := Nat.mem_properDivisors.2
     ⟨hm, lt_of_le_of_ne (Nat.divisor_le (Nat.mem_divisors.2 ⟨hm, hpos.ne'⟩)) hdiff⟩
@@ -519,7 +519,7 @@ set_option linter.uppercaseLean3 false in
 /-- If there is a primitive `n`-th root of unity in `K`, then
 `cyclotomic n K = ∏ μ in primitiveRoots n K, (X - C μ)`. In particular,
 `cyclotomic n K = cyclotomic' n K` -/
-theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type _} [CommRing K] [IsDomain K] {ζ : K}
+theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type*} [CommRing K] [IsDomain K] {ζ : K}
     {n : ℕ} (hz : IsPrimitiveRoot ζ n) : cyclotomic n K = ∏ μ in primitiveRoots n K, (X - C μ) := by
   rw [← cyclotomic']
   induction' n using Nat.strong_induction_on with k hk generalizing ζ
@@ -535,7 +535,7 @@ theorem cyclotomic_eq_prod_X_sub_primitiveRoots {K : Type _} [CommRing K] [IsDom
 set_option linter.uppercaseLean3 false in
 #align polynomial.cyclotomic_eq_prod_X_sub_primitive_roots Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots
 
-theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P : R[X]) :
+theorem eq_cyclotomic_iff {R : Type*} [CommRing R] {n : ℕ} (hpos : 0 < n) (P : R[X]) :
     P = cyclotomic n R ↔
     (P * ∏ i in Nat.properDivisors n, Polynomial.cyclotomic i R) = X ^ n - 1 := by
   nontriviality R
@@ -555,7 +555,7 @@ theorem eq_cyclotomic_iff {R : Type _} [CommRing R] {n : ℕ} (hpos : 0 < n) (P
 
 /-- If `p ^ k` is a prime power, then
 `cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ (p ^ n)) ^ i`. -/
-theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (hp : p.Prime) :
+theorem cyclotomic_prime_pow_eq_geom_sum {R : Type*} [CommRing R] {p n : ℕ} (hp : p.Prime) :
     cyclotomic (p ^ (n + 1)) R = ∑ i in Finset.range p, (X ^ p ^ n) ^ i := by
   have : ∀ m, (cyclotomic (p ^ (m + 1)) R = ∑ i in Finset.range p, (X ^ p ^ m) ^ i) ↔
       ((∑ i in Finset.range p, (X ^ p ^ m) ^ i) *
@@ -573,7 +573,7 @@ theorem cyclotomic_prime_pow_eq_geom_sum {R : Type _} [CommRing R] {p n : ℕ} (
   rw [mul_comm _ (∑ i in _, _), n_ih, geom_sum_mul, sub_left_inj, ← pow_mul, pow_add, pow_one]
 #align polynomial.cyclotomic_prime_pow_eq_geom_sum Polynomial.cyclotomic_prime_pow_eq_geom_sum
 
-theorem cyclotomic_prime_pow_mul_X_pow_sub_one (R : Type _) [CommRing R] (p k : ℕ)
+theorem cyclotomic_prime_pow_mul_X_pow_sub_one (R : Type*) [CommRing R] (p k : ℕ)
     [hn : Fact (Nat.Prime p)] :
     cyclotomic (p ^ (k + 1)) R * (X ^ p ^ k - 1) = X ^ p ^ (k + 1) - 1 := by
   rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, ← pow_mul, pow_succ, mul_comm]
@@ -581,7 +581,7 @@ set_option linter.uppercaseLean3 false in
 #align polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one
 
 /-- The constant term of `cyclotomic n R` is `1` if `2 ≤ n`. -/
-theorem cyclotomic_coeff_zero (R : Type _) [CommRing R] {n : ℕ} (hn : 1 < n) :
+theorem cyclotomic_coeff_zero (R : Type*) [CommRing R] {n : ℕ} (hn : 1 < n) :
     (cyclotomic n R).coeff 0 = 1 := by
   induction' n using Nat.strong_induction_on with n hi
   have hprod : (∏ i in Nat.properDivisors n, (Polynomial.cyclotomic i R).coeff 0) = -1 := by

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.cyclotomic.basic
-! leanprover-community/mathlib commit 7fdeecc0d03cd40f7a165e6cf00a4d2286db599f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.NeZero
 import Mathlib.Algebra.Polynomial.BigOperators
@@ -19,6 +14,8 @@ import Mathlib.FieldTheory.Separable
 import Mathlib.NumberTheory.ArithmeticFunction
 import Mathlib.RingTheory.RootsOfUnity.Basic
 
+#align_import ring_theory.polynomial.cyclotomic.basic from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
+
 /-!
 # Cyclotomic polynomials.
 

• Also remove most superfluous parentheses around big operators (∑, ∏ and variants).
• roughly the used regex: ([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤]) replaced by $1 $2 $3

@@ -170,7 +170,7 @@ set_option linter.uppercaseLean3 false in
 `∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1`. -/
 theorem prod_cyclotomic'_eq_X_pow_sub_one {K : Type _} [CommRing K] [IsDomain K] {ζ : K} {n : ℕ}
     (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
-    (∏ i in Nat.divisors n, cyclotomic' i K) = X ^ n - 1 := by
+    ∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1 := by
   have hd : (n.divisors : Set ℕ).PairwiseDisjoint fun k => primitiveRoots k K :=
     fun x _ y _ hne => IsPrimitiveRoot.disjoint hne
   simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← Finset.prod_biUnion hd,
@@ -363,8 +363,8 @@ open Finset
 
 /-- `∏ i in Nat.divisors n, cyclotomic i R = X ^ n - 1`. -/
 theorem prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type _) [CommRing R] :
-    (∏ i in Nat.divisors n, cyclotomic i R) = X ^ n - 1 := by
-  have integer : (∏ i in Nat.divisors n, cyclotomic i ℤ) = X ^ n - 1 := by
+    ∏ i in Nat.divisors n, cyclotomic i R = X ^ n - 1 := by
+  have integer : ∏ i in Nat.divisors n, cyclotomic i ℤ = X ^ n - 1 := by
     apply map_injective (Int.castRingHom ℂ) Int.cast_injective
     simp only [Polynomial.map_prod, int_cyclotomic_spec, Polynomial.map_pow, map_X,
       Polynomial.map_one, Polynomial.map_sub]
@@ -387,7 +387,7 @@ set_option linter.uppercaseLean3 false in
 #align polynomial.cyclotomic.dvd_X_pow_sub_one Polynomial.cyclotomic.dvd_X_pow_sub_one
 
 theorem prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [CommRing R] :
-    (∏ i in n.divisors.erase 1, cyclotomic i R) = ∑ i in Finset.range n, X ^ i := by
+    ∏ i in n.divisors.erase 1, cyclotomic i R = ∑ i in Finset.range n, X ^ i := by
   suffices (∏ i in n.divisors.erase 1, cyclotomic i ℤ) = ∑ i in Finset.range n, X ^ i by
     simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
       Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this