analysis.complex.polynomial | Mathlib porting status

Changes since the initial port

The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.

Changes in mathlib3

17ef379e

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

660b3a2d

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -36,7 +36,7 @@ theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z :
   · obtain rfl : f = C c⁻¹ := Polynomial.funext fun z => by rw [eval_C, ← hc z, inv_inv]
     exact degree_C_le
   · obtain ⟨z₀, h₀⟩ := f.exists_forall_norm_le
-    simp only [isBounded_iff_forall_norm_le, Set.forall_range_iff, norm_inv]
+    simp only [isBounded_iff_forall_norm_le, Set.forall_mem_range, norm_inv]
     exact ⟨‖eval z₀ f‖⁻¹, fun z => inv_le_inv_of_le (norm_pos_iff.2 <| hf z₀) (h₀ z)⟩
 #align complex.exists_root Complex.exists_root
 -/

660b3a2d

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,8 +3,8 @@ Copyright (c) 2019 Chris Hughes All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Junyan Xu
 -/
-import Mathbin.Analysis.Complex.Liouville
-import Mathbin.FieldTheory.IsAlgClosed.Basic
+import Analysis.Complex.Liouville
+import FieldTheory.IsAlgClosed.Basic
 
 #align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"660b3a2db3522fa0db036e569dc995a615c4c848"

660b3a2d

bump to nightly-2023-09-20-06

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

e28e6964

Diff

@@ -36,7 +36,7 @@ theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z :
   · obtain rfl : f = C c⁻¹ := Polynomial.funext fun z => by rw [eval_C, ← hc z, inv_inv]
     exact degree_C_le
   · obtain ⟨z₀, h₀⟩ := f.exists_forall_norm_le
-    simp only [bounded_iff_forall_norm_le, Set.forall_range_iff, norm_inv]
+    simp only [isBounded_iff_forall_norm_le, Set.forall_range_iff, norm_inv]
     exact ⟨‖eval z₀ f‖⁻¹, fun z => inv_le_inv_of_le (norm_pos_iff.2 <| hf z₀) (h₀ z)⟩
 #align complex.exists_root Complex.exists_root
 -/

660b3a2d

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,15 +2,12 @@
 Copyright (c) 2019 Chris Hughes All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Junyan Xu
-
-! This file was ported from Lean 3 source module analysis.complex.polynomial
-! leanprover-community/mathlib commit 660b3a2db3522fa0db036e569dc995a615c4c848
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Complex.Liouville
 import Mathbin.FieldTheory.IsAlgClosed.Basic
 
+#align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"660b3a2db3522fa0db036e569dc995a615c4c848"
+
 /-!
 # The fundamental theorem of algebra

660b3a2d

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -44,9 +44,11 @@ theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z :
 #align complex.exists_root Complex.exists_root
 -/
 
+#print Complex.isAlgClosed /-
 instance isAlgClosed : IsAlgClosed ℂ :=
   IsAlgClosed.of_exists_root _ fun p _ hp => Complex.exists_root <| degree_pos_of_irreducible hp
 #align complex.is_alg_closed Complex.isAlgClosed
+-/
 
 end Complex

660b3a2d

bump to nightly-2023-06-12-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f

879d466d

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Junyan Xu
 
 ! This file was ported from Lean 3 source module analysis.complex.polynomial
-! leanprover-community/mathlib commit 17ef379e997badd73e5eabb4d38f11919ab3c4b3
+! leanprover-community/mathlib commit 660b3a2db3522fa0db036e569dc995a615c4c848
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.FieldTheory.IsAlgClosed.Basic
 /-!
 # The fundamental theorem of algebra
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file proves that every nonconstant complex polynomial has a root using Liouville's theorem.
 
 As a consequence, the complex numbers are algebraically closed.

17ef379e

bump to nightly-2023-06-10-07

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

3fdc57bc

Diff

@@ -26,6 +26,7 @@ open scoped Polynomial
 
 namespace Complex
 
+#print Complex.exists_root /-
 /-- **Fundamental theorem of algebra**: every non constant complex polynomial
   has a root -/
 theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z :=
@@ -38,6 +39,7 @@ theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z :
     simp only [bounded_iff_forall_norm_le, Set.forall_range_iff, norm_inv]
     exact ⟨‖eval z₀ f‖⁻¹, fun z => inv_le_inv_of_le (norm_pos_iff.2 <| hf z₀) (h₀ z)⟩
 #align complex.exists_root Complex.exists_root
+-/
 
 instance isAlgClosed : IsAlgClosed ℂ :=
   IsAlgClosed.of_exists_root _ fun p _ hp => Complex.exists_root <| degree_pos_of_irreducible hp

17ef379e

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -22,7 +22,7 @@ As a consequence, the complex numbers are algebraically closed.
 
 open Polynomial
 
-open Polynomial
+open scoped Polynomial
 
 namespace Complex

17ef379e

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

17ef379e

chore: adapt to multiple goal linter 1 (#12338)

A PR accompanying #12339.

Zulip discussion

45f93f3f

Diff

@@ -77,8 +77,8 @@ theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
   rw [← Finset.card_image_of_injective _ Subtype.coe_injective, ←
     Finset.card_image_of_injective _ inj]
   let a : Finset ℂ := ?_
-  let b : Finset ℂ := ?_
-  let c : Finset ℂ := ?_
+  on_goal 1 => let b : Finset ℂ := ?_
+  on_goal 1 => let c : Finset ℂ := ?_
   -- Porting note: was
   --   change a.card = b.card + c.card
   suffices a.card = b.card + c.card by exact this
@@ -91,7 +91,10 @@ theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
     · rintro ⟨w, hw, rfl⟩
       exact ⟨by rw [aeval_algHom_apply, hw, AlgHom.map_zero], rfl⟩
     · rintro ⟨hz1, hz2⟩
-      have key : IsScalarTower.toAlgHom ℚ ℝ ℂ z.re = z := by ext; rfl; rw [hz2]; rfl
+      have key : IsScalarTower.toAlgHom ℚ ℝ ℂ z.re = z := by
+        ext
+        · rfl
+        · rw [hz2]; rfl
       exact ⟨z.re, inj (by rwa [← aeval_algHom_apply, key, AlgHom.map_zero]), key⟩
   have hc0 :
     ∀ w : p.rootSet ℂ, galActionHom p ℂ (restrict p ℂ (Complex.conjAe.restrictScalars ℚ)) w = w ↔

17ef379e

chore: Rename IsROrC to RCLike (#10819)

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

43618f93

Diff

@@ -205,7 +205,7 @@ lemma Irreducible.degree_le_two {p : ℝ[X]} (hp : Irreducible p) : degree p ≤
   cases eq_or_ne z.im 0 with
   | inl hz0 =>
     lift z to ℝ using hz0
-    erw [aeval_ofReal, IsROrC.ofReal_eq_zero] at hz
+    erw [aeval_ofReal, RCLike.ofReal_eq_zero] at hz
     exact (degree_eq_one_of_irreducible_of_root hp hz).trans_le one_le_two
   | inr hz0 =>
     obtain ⟨q, rfl⟩ := p.quadratic_dvd_of_aeval_eq_zero_im_ne_zero hz hz0

17ef379e

chore: move Mathlib to v4.7.0-rc1 (#11162)

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

fa48894a

Diff

@@ -86,7 +86,7 @@ theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
     intro z; rw [Set.mem_toFinset, mem_rootSet_of_ne hp]
   have hb : ∀ z : ℂ, z ∈ b ↔ aeval z p = 0 ∧ z.im = 0 := by
     intro z
-    simp_rw [Finset.mem_image, Set.mem_toFinset, mem_rootSet_of_ne hp]
+    simp_rw [b, Finset.mem_image, Set.mem_toFinset, mem_rootSet_of_ne hp]
     constructor
     · rintro ⟨w, hw, rfl⟩
       exact ⟨by rw [aeval_algHom_apply, hw, AlgHom.map_zero], rfl⟩
@@ -101,7 +101,7 @@ theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
     exact Complex.conj_eq_iff_im
   have hc : ∀ z : ℂ, z ∈ c ↔ aeval z p = 0 ∧ z.im ≠ 0 := by
     intro z
-    simp_rw [Finset.mem_image]
+    simp_rw [c, Finset.mem_image]
     constructor
     · rintro ⟨w, hw, rfl⟩
       exact ⟨(mem_rootSet.mp w.2).2, mt (hc0 w).mpr (Equiv.Perm.mem_support.mp hw)⟩

17ef379e

fix: rename Irreducible.nat_degree_le_two (#10687)

89162a46

Diff

@@ -219,5 +219,8 @@ lemma Irreducible.degree_le_two {p : ℝ[X]} (hp : Irreducible p) : degree p ≤
     rwa [isUnit_iff_degree_eq_zero.1 hq, add_zero]
 
 /-- An irreducible real polynomial has natural degree at most two. -/
-lemma Irreducible.nat_degree_le_two {p : ℝ[X]} (hp : Irreducible p) : natDegree p ≤ 2 :=
+lemma Irreducible.natDegree_le_two {p : ℝ[X]} (hp : Irreducible p) : natDegree p ≤ 2 :=
   natDegree_le_iff_degree_le.2 hp.degree_le_two
+
+@[deprecated] -- 2024-02-18
+alias Irreducible.nat_degree_le_two := Irreducible.natDegree_le_two

17ef379e

perf: speed up galActionHom_bijective_of_prime_degree (#10608)

This takes the proof from approximately 3 seconds to half a second on my laptop.

convert spent a fair amount of time dealing with the equality of Fintype.card calls with the same type but different Fintype instances. This sidesteps the issue by translating to Nat.card, which doesn't have the instance argument.

e8f18827

Diff

@@ -128,8 +128,6 @@ theorem galActionHom_bijective_of_prime_degree {p : ℚ[X]} (p_irr : Irreducible
     rw [Multiset.toFinset_card_of_nodup, ← natDegree_eq_card_roots]
     · exact IsAlgClosed.splits_codomain p
     · exact nodup_roots ((separable_map (algebraMap ℚ ℂ)).mpr p_irr.separable)
-  have h2 : Fintype.card p.Gal = Fintype.card (galActionHom p ℂ).range :=
-    Fintype.card_congr (MonoidHom.ofInjective (galActionHom_injective p ℂ)).toEquiv
   let conj' := restrict p ℂ (Complex.conjAe.restrictScalars ℚ)
   refine'
     ⟨galActionHom_injective p ℂ, fun x =>
@@ -138,8 +136,9 @@ theorem galActionHom_bijective_of_prime_degree {p : ℚ[X]} (p_irr : Irreducible
   apply Equiv.Perm.subgroup_eq_top_of_swap_mem
   · rwa [h1]
   · rw [h1]
-    convert prime_degree_dvd_card p_irr p_deg using 1
-    convert h2.symm
+    simpa only [Fintype.card_eq_nat_card,
+      Nat.card_congr (MonoidHom.ofInjective (galActionHom_injective p ℂ)).toEquiv.symm]
+      using prime_degree_dvd_card p_irr p_deg
   · exact ⟨conj', rfl⟩
   · rw [← Equiv.Perm.card_support_eq_two]
     apply Nat.add_left_cancel

17ef379e

chore(FieldTheory/PolynomialGaloisGroup): move lemmas, reduce imports (#9886)

#dependencies: Mathlib.FieldTheory.PolynomialGaloisGroup: 1826->1323 Mathlib.Analysis.Complex.Polynomial: 1811->1826

This needed one small change to a moved proof because conj is not allowed as an identifier when the ComplexConjugate locale is open.

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

8257db38

Diff

@@ -3,10 +3,9 @@ Copyright (c) 2019 Chris Hughes All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Junyan Xu, Yury Kudryashov
 -/
-import Mathlib.Algebra.CharZero.Infinite
 import Mathlib.Analysis.Complex.Liouville
 import Mathlib.Analysis.Calculus.Deriv.Polynomial
-import Mathlib.FieldTheory.IsAlgClosed.Basic
+import Mathlib.FieldTheory.PolynomialGaloisGroup
 import Mathlib.Topology.Algebra.Polynomial
 
 #align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
@@ -18,6 +17,9 @@ This file proves that every nonconstant complex polynomial has a root using Liou
 
 As a consequence, the complex numbers are algebraically closed.
 
+We also provide some specific results about the Galois groups of ℚ-polynomials with specific numbers
+of non-real roots.
+
 We also show that an irreducible real polynomial has degree at most two.
 -/
 
@@ -49,6 +51,134 @@ instance isAlgClosed : IsAlgClosed ℂ :=
 
 end Complex
 
+namespace Polynomial.Gal
+
+section Rationals
+
+theorem splits_ℚ_ℂ {p : ℚ[X]} : Fact (p.Splits (algebraMap ℚ ℂ)) :=
+  ⟨IsAlgClosed.splits_codomain p⟩
+#align polynomial.gal.splits_ℚ_ℂ Polynomial.Gal.splits_ℚ_ℂ
+
+attribute [local instance] splits_ℚ_ℂ
+attribute [local ext] Complex.ext
+
+/-- The number of complex roots equals the number of real roots plus
+    the number of roots not fixed by complex conjugation (i.e. with some imaginary component). -/
+theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
+    (p.rootSet ℂ).toFinset.card =
+      (p.rootSet ℝ).toFinset.card +
+        (galActionHom p ℂ (restrict p ℂ
+        (AlgEquiv.restrictScalars ℚ Complex.conjAe))).support.card := by
+  by_cases hp : p = 0
+  · haveI : IsEmpty (p.rootSet ℂ) := by rw [hp, rootSet_zero]; infer_instance
+    simp_rw [(galActionHom p ℂ _).support.eq_empty_of_isEmpty, hp, rootSet_zero,
+      Set.toFinset_empty, Finset.card_empty]
+  have inj : Function.Injective (IsScalarTower.toAlgHom ℚ ℝ ℂ) := (algebraMap ℝ ℂ).injective
+  rw [← Finset.card_image_of_injective _ Subtype.coe_injective, ←
+    Finset.card_image_of_injective _ inj]
+  let a : Finset ℂ := ?_
+  let b : Finset ℂ := ?_
+  let c : Finset ℂ := ?_
+  -- Porting note: was
+  --   change a.card = b.card + c.card
+  suffices a.card = b.card + c.card by exact this
+  have ha : ∀ z : ℂ, z ∈ a ↔ aeval z p = 0 := by
+    intro z; rw [Set.mem_toFinset, mem_rootSet_of_ne hp]
+  have hb : ∀ z : ℂ, z ∈ b ↔ aeval z p = 0 ∧ z.im = 0 := by
+    intro z
+    simp_rw [Finset.mem_image, Set.mem_toFinset, mem_rootSet_of_ne hp]
+    constructor
+    · rintro ⟨w, hw, rfl⟩
+      exact ⟨by rw [aeval_algHom_apply, hw, AlgHom.map_zero], rfl⟩
+    · rintro ⟨hz1, hz2⟩
+      have key : IsScalarTower.toAlgHom ℚ ℝ ℂ z.re = z := by ext; rfl; rw [hz2]; rfl
+      exact ⟨z.re, inj (by rwa [← aeval_algHom_apply, key, AlgHom.map_zero]), key⟩
+  have hc0 :
+    ∀ w : p.rootSet ℂ, galActionHom p ℂ (restrict p ℂ (Complex.conjAe.restrictScalars ℚ)) w = w ↔
+        w.val.im = 0 := by
+    intro w
+    rw [Subtype.ext_iff, galActionHom_restrict]
+    exact Complex.conj_eq_iff_im
+  have hc : ∀ z : ℂ, z ∈ c ↔ aeval z p = 0 ∧ z.im ≠ 0 := by
+    intro z
+    simp_rw [Finset.mem_image]
+    constructor
+    · rintro ⟨w, hw, rfl⟩
+      exact ⟨(mem_rootSet.mp w.2).2, mt (hc0 w).mpr (Equiv.Perm.mem_support.mp hw)⟩
+    · rintro ⟨hz1, hz2⟩
+      exact ⟨⟨z, mem_rootSet.mpr ⟨hp, hz1⟩⟩, Equiv.Perm.mem_support.mpr (mt (hc0 _).mp hz2), rfl⟩
+  rw [← Finset.card_union_of_disjoint]
+  · apply congr_arg Finset.card
+    simp_rw [Finset.ext_iff, Finset.mem_union, ha, hb, hc]
+    tauto
+  · rw [Finset.disjoint_left]
+    intro z
+    rw [hb, hc]
+    tauto
+#align polynomial.gal.card_complex_roots_eq_card_real_add_card_not_gal_inv Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv
+
+/-- An irreducible polynomial of prime degree with two non-real roots has full Galois group. -/
+theorem galActionHom_bijective_of_prime_degree {p : ℚ[X]} (p_irr : Irreducible p)
+    (p_deg : p.natDegree.Prime)
+    (p_roots : Fintype.card (p.rootSet ℂ) = Fintype.card (p.rootSet ℝ) + 2) :
+    Function.Bijective (galActionHom p ℂ) := by
+  classical
+  have h1 : Fintype.card (p.rootSet ℂ) = p.natDegree := by
+    simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]
+    rw [Multiset.toFinset_card_of_nodup, ← natDegree_eq_card_roots]
+    · exact IsAlgClosed.splits_codomain p
+    · exact nodup_roots ((separable_map (algebraMap ℚ ℂ)).mpr p_irr.separable)
+  have h2 : Fintype.card p.Gal = Fintype.card (galActionHom p ℂ).range :=
+    Fintype.card_congr (MonoidHom.ofInjective (galActionHom_injective p ℂ)).toEquiv
+  let conj' := restrict p ℂ (Complex.conjAe.restrictScalars ℚ)
+  refine'
+    ⟨galActionHom_injective p ℂ, fun x =>
+      (congr_arg (Membership.mem x) (show (galActionHom p ℂ).range = ⊤ from _)).mpr
+        (Subgroup.mem_top x)⟩
+  apply Equiv.Perm.subgroup_eq_top_of_swap_mem
+  · rwa [h1]
+  · rw [h1]
+    convert prime_degree_dvd_card p_irr p_deg using 1
+    convert h2.symm
+  · exact ⟨conj', rfl⟩
+  · rw [← Equiv.Perm.card_support_eq_two]
+    apply Nat.add_left_cancel
+    rw [← p_roots, ← Set.toFinset_card (rootSet p ℝ), ← Set.toFinset_card (rootSet p ℂ)]
+    exact (card_complex_roots_eq_card_real_add_card_not_gal_inv p).symm
+#align polynomial.gal.gal_action_hom_bijective_of_prime_degree Polynomial.Gal.galActionHom_bijective_of_prime_degree
+
+/-- An irreducible polynomial of prime degree with 1-3 non-real roots has full Galois group. -/
+theorem galActionHom_bijective_of_prime_degree' {p : ℚ[X]} (p_irr : Irreducible p)
+    (p_deg : p.natDegree.Prime)
+    (p_roots1 : Fintype.card (p.rootSet ℝ) + 1 ≤ Fintype.card (p.rootSet ℂ))
+    (p_roots2 : Fintype.card (p.rootSet ℂ) ≤ Fintype.card (p.rootSet ℝ) + 3) :
+    Function.Bijective (galActionHom p ℂ) := by
+  apply galActionHom_bijective_of_prime_degree p_irr p_deg
+  let n := (galActionHom p ℂ (restrict p ℂ (Complex.conjAe.restrictScalars ℚ))).support.card
+  have hn : 2 ∣ n :=
+    Equiv.Perm.two_dvd_card_support
+      (by
+         rw [← MonoidHom.map_pow, ← MonoidHom.map_pow,
+          show AlgEquiv.restrictScalars ℚ Complex.conjAe ^ 2 = 1 from
+            AlgEquiv.ext Complex.conj_conj,
+          MonoidHom.map_one, MonoidHom.map_one])
+  have key := card_complex_roots_eq_card_real_add_card_not_gal_inv p
+  simp_rw [Set.toFinset_card] at key
+  rw [key, add_le_add_iff_left] at p_roots1 p_roots2
+  rw [key, add_right_inj]
+  suffices ∀ m : ℕ, 2 ∣ m → 1 ≤ m → m ≤ 3 → m = 2 by exact this n hn p_roots1 p_roots2
+  rintro m ⟨k, rfl⟩ h2 h3
+  exact le_antisymm
+      (Nat.lt_succ_iff.mp
+        (lt_of_le_of_ne h3 (show 2 * k ≠ 2 * 1 + 1 from Nat.two_mul_ne_two_mul_add_one)))
+      (Nat.succ_le_iff.mpr
+        (lt_of_le_of_ne h2 (show 2 * 0 + 1 ≠ 2 * k from Nat.two_mul_ne_two_mul_add_one.symm)))
+#align polynomial.gal.gal_action_hom_bijective_of_prime_degree' Polynomial.Gal.galActionHom_bijective_of_prime_degree'
+
+end Rationals
+
+end Polynomial.Gal
+
 lemma Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero (p : ℝ[X]) {z : ℂ} (h0 : aeval z p = 0)
     (hz : z.im ≠ 0) : (X - C ((starRingEnd ℂ) z)) * (X - C z) ∣ map (algebraMap ℝ ℂ) p := by
   apply IsCoprime.mul_dvd

17ef379e

feat: an irreducible real polynomial has degree ≤2 (#10431)

30c9985e

Diff

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Chris Hughes All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Junyan Xu
+Authors: Chris Hughes, Junyan Xu, Yury Kudryashov
 -/
 import Mathlib.Algebra.CharZero.Infinite
 import Mathlib.Analysis.Complex.Liouville
@@ -17,11 +17,13 @@ import Mathlib.Topology.Algebra.Polynomial
 This file proves that every nonconstant complex polynomial has a root using Liouville's theorem.
 
 As a consequence, the complex numbers are algebraically closed.
+
+We also show that an irreducible real polynomial has degree at most two.
 -/
 
-open Polynomial Bornology
+open Polynomial Bornology Complex
 
-open scoped Polynomial
+open scoped ComplexConjugate
 
 namespace Complex
 
@@ -46,3 +48,47 @@ instance isAlgClosed : IsAlgClosed ℂ :=
 #align complex.is_alg_closed Complex.isAlgClosed
 
 end Complex
+
+lemma Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero (p : ℝ[X]) {z : ℂ} (h0 : aeval z p = 0)
+    (hz : z.im ≠ 0) : (X - C ((starRingEnd ℂ) z)) * (X - C z) ∣ map (algebraMap ℝ ℂ) p := by
+  apply IsCoprime.mul_dvd
+  · exact isCoprime_X_sub_C_of_isUnit_sub <| .mk0 _ <| sub_ne_zero.2 <| mt conj_eq_iff_im.1 hz
+  · simpa [dvd_iff_isRoot, aeval_conj]
+  · simpa [dvd_iff_isRoot]
+
+/-- If `z` is a non-real complex root of a real polynomial,
+then `p` is divisible by a quadratic polynomial. -/
+lemma Polynomial.quadratic_dvd_of_aeval_eq_zero_im_ne_zero (p : ℝ[X]) {z : ℂ} (h0 : aeval z p = 0)
+    (hz : z.im ≠ 0) : X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2) ∣ p := by
+  rw [← map_dvd_map' (algebraMap ℝ ℂ)]
+  convert p.mul_star_dvd_of_aeval_eq_zero_im_ne_zero h0 hz
+  calc
+    map (algebraMap ℝ ℂ) (X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2))
+    _ = X ^ 2 - C (↑(2 * z.re) : ℂ) * X + C (‖z‖ ^ 2 : ℂ) := by simp
+    _ = (X - C (conj z)) * (X - C z) := by
+      rw [← add_conj, map_add, ← mul_conj', map_mul]
+      ring
+
+/-- An irreducible real polynomial has degree at most two. -/
+lemma Irreducible.degree_le_two {p : ℝ[X]} (hp : Irreducible p) : degree p ≤ 2 := by
+  obtain ⟨z, hz⟩ : ∃ z : ℂ, aeval z p = 0 :=
+    IsAlgClosed.exists_aeval_eq_zero _ p (degree_pos_of_irreducible hp).ne'
+  cases eq_or_ne z.im 0 with
+  | inl hz0 =>
+    lift z to ℝ using hz0
+    erw [aeval_ofReal, IsROrC.ofReal_eq_zero] at hz
+    exact (degree_eq_one_of_irreducible_of_root hp hz).trans_le one_le_two
+  | inr hz0 =>
+    obtain ⟨q, rfl⟩ := p.quadratic_dvd_of_aeval_eq_zero_im_ne_zero hz hz0
+    have hd : degree (X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2)) = 2 := by
+      compute_degree!
+    have hq : IsUnit q := by
+      refine (of_irreducible_mul hp).resolve_left (mt isUnit_iff_degree_eq_zero.1 ?_)
+      rw [hd]
+      exact two_ne_zero
+    refine (degree_mul_le _ _).trans_eq ?_
+    rwa [isUnit_iff_degree_eq_zero.1 hq, add_zero]
+
+/-- An irreducible real polynomial has natural degree at most two. -/
+lemma Irreducible.nat_degree_le_two {p : ℝ[X]} (hp : Irreducible p) : natDegree p ≤ 2 :=
+  natDegree_le_iff_degree_le.2 hp.degree_le_two

17ef379e

chore: reduce imports (#9830)

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>

f4c4ca61

Diff

@@ -3,9 +3,10 @@ Copyright (c) 2019 Chris Hughes All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Junyan Xu
 -/
+import Mathlib.Algebra.CharZero.Infinite
 import Mathlib.Analysis.Complex.Liouville
-import Mathlib.FieldTheory.IsAlgClosed.Basic
 import Mathlib.Analysis.Calculus.Deriv.Polynomial
+import Mathlib.FieldTheory.IsAlgClosed.Basic
 import Mathlib.Topology.Algebra.Polynomial
 
 #align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"

17ef379e

chore: rename by_contra' to by_contra! (#8797)

To fit with the "please try harder" convention of ! tactics.

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

738b1a97

Diff

@@ -27,7 +27,7 @@ namespace Complex
 /-- **Fundamental theorem of algebra**: every non constant complex polynomial
   has a root -/
 theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z := by
-  by_contra' hf'
+  by_contra! hf'
   /- Since `f` has no roots, `f⁻¹` is differentiable. And since `f` is a polynomial, it tends to
   infinity at infinity, thus `f⁻¹` tends to zero at infinity. By Liouville's theorem, `f⁻¹ = 0`. -/
   have (z : ℂ) : (f.eval z)⁻¹ = 0 :=

17ef379e

feat: conceptualize the proof of the fundamental theorem of algebra (#8245)

Uses the corollary Differentiable.apply_eq_of_tendsto_cocompact of Liouville's theorem to refactor the proof of the Fundamental theorem of Algebra in order to make it slightly more conceptual.

depends on: #8234
depends on: #8244

b6a9eb9c

Diff

@@ -27,14 +27,17 @@ namespace Complex
 /-- **Fundamental theorem of algebra**: every non constant complex polynomial
   has a root -/
 theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z := by
-  contrapose! hf
-  have : IsBounded (Set.range (eval · f)⁻¹)
-  · obtain ⟨z₀, h₀⟩ := f.exists_forall_norm_le
-    simp only [Pi.inv_apply, isBounded_iff_forall_norm_le, Set.forall_range_iff, norm_inv]
-    exact ⟨‖eval z₀ f‖⁻¹, fun z => inv_le_inv_of_le (norm_pos_iff.2 <| hf z₀) (h₀ z)⟩
-  obtain ⟨c, hc⟩ := (f.differentiable.inv hf).exists_const_forall_eq_of_bounded this
-  · obtain rfl : f = C c⁻¹ := Polynomial.funext fun z => by rw [eval_C, ← hc z, inv_inv]
-    exact degree_C_le
+  by_contra' hf'
+  /- Since `f` has no roots, `f⁻¹` is differentiable. And since `f` is a polynomial, it tends to
+  infinity at infinity, thus `f⁻¹` tends to zero at infinity. By Liouville's theorem, `f⁻¹ = 0`. -/
+  have (z : ℂ) : (f.eval z)⁻¹ = 0 :=
+    (f.differentiable.inv hf').apply_eq_of_tendsto_cocompact z <|
+      Metric.cobounded_eq_cocompact (α := ℂ) ▸ (Filter.tendsto_inv₀_cobounded.comp <| by
+        simpa only [tendsto_norm_atTop_iff_cobounded]
+          using f.tendsto_norm_atTop hf tendsto_norm_cobounded_atTop)
+  -- Thus `f = 0`, contradicting the fact that `0 < degree f`.
+  obtain rfl : f = C 0 := Polynomial.funext fun z ↦ inv_injective <| by simp [this]
+  simp at hf
 #align complex.exists_root Complex.exists_root
 
 instance isAlgClosed : IsAlgClosed ℂ :=

17ef379e

refactor(Topology/MetricSpace): remove Metric.Bounded (#7240)

Use Bornology.IsBounded instead.

98e78f90

Diff

@@ -18,7 +18,7 @@ This file proves that every nonconstant complex polynomial has a root using Liou
 As a consequence, the complex numbers are algebraically closed.
 -/
 
-open Polynomial
+open Polynomial Bornology
 
 open scoped Polynomial
 
@@ -28,9 +28,9 @@ namespace Complex
   has a root -/
 theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z := by
   contrapose! hf
-  have : Metric.Bounded (Set.range (eval · f)⁻¹)
+  have : IsBounded (Set.range (eval · f)⁻¹)
   · obtain ⟨z₀, h₀⟩ := f.exists_forall_norm_le
-    simp only [Pi.inv_apply, bounded_iff_forall_norm_le, Set.forall_range_iff, norm_inv]
+    simp only [Pi.inv_apply, isBounded_iff_forall_norm_le, Set.forall_range_iff, norm_inv]
     exact ⟨‖eval z₀ f‖⁻¹, fun z => inv_le_inv_of_le (norm_pos_iff.2 <| hf z₀) (h₀ z)⟩
   obtain ⟨c, hc⟩ := (f.differentiable.inv hf).exists_const_forall_eq_of_bounded this
   · obtain rfl : f = C c⁻¹ := Polynomial.funext fun z => by rw [eval_C, ← hc z, inv_inv]

17ef379e

refactor: split Analysis.Calculus.LocalExtr (#5944)

Also make f/f' arguments implicit in all versions of Rolle's Theorem.

Fixes #4830

API changes

exists_Ioo_extr_on_Icc:
- generalize from functions f : ℝ → ℝ to functions from a conditionally complete linear order to a linear order.
- make f implicit;
exists_local_extr_Ioo:
- rename to exists_isLocalExtr_Ioo;
- generalize as above;
- make f implicit;
exists_isExtrOn_Ioo_of_tendsto, exists_isLocalExtr_Ioo_of_tendsto: new lemmas extracted from the proof of exists_hasDerivAt_eq_zero';
exists_hasDerivAt_eq_zero, exists_hasDerivAt_eq_zero': make f and f' implicit;
exists_deriv_eq_zero, exists_deriv_eq_zero': make f implicit.

eec218b4

Diff

@@ -5,6 +5,8 @@ Authors: Chris Hughes, Junyan Xu
 -/
 import Mathlib.Analysis.Complex.Liouville
 import Mathlib.FieldTheory.IsAlgClosed.Basic
+import Mathlib.Analysis.Calculus.Deriv.Polynomial
+import Mathlib.Topology.Algebra.Polynomial
 
 #align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"

17ef379e

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

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

89aa3a3b

Diff

@@ -2,15 +2,12 @@
 Copyright (c) 2019 Chris Hughes All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Junyan Xu
-
-! This file was ported from Lean 3 source module analysis.complex.polynomial
-! leanprover-community/mathlib commit 17ef379e997badd73e5eabb4d38f11919ab3c4b3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Complex.Liouville
 import Mathlib.FieldTheory.IsAlgClosed.Basic
 
+#align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
+
 /-!
 # The fundamental theorem of algebra

17ef379e

feat: port Analysis.Complex.Polynomial (#4931)

e3d3b979

`analysis.complex.polynomial` ⟷ `Mathlib.Analysis.Complex.Polynomial`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

API changes

Dependencies 12 + 1126

analysis.complex.polynomial ⟷ Mathlib.Analysis.Complex.Polynomial

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

API changes

Dependencies 12 + 1126

`analysis.complex.polynomial` ⟷ `Mathlib.Analysis.Complex.Polynomial`