Split polynomials

A polynomial f : R[X] splits if it is a product of constant and monic linear polynomials.

Main definitions

• Polynomial.Splits f: A predicate on a polynomial f saying that f is a product of constant and monic linear polynomials.

def Polynomial.Splits {R : Type u_1} [Semiring R] (f : Polynomial R) : Prop

A polynomial Splits if it is a product of constant and monic linear polynomials. This will eventually replace Polynomial.Splits.

Equations

• f.Splits = (f ∈ Submonoid.closure ({x : Polynomial R | ∃ (a : R), Polynomial.C a = x} ∪ {x : Polynomial R | ∃ (a : R), Polynomial.X + Polynomial.C a = x}))

@[simp]

theorem Polynomial.Splits.C {R : Type u_1} [Semiring R] (a : R) : (C a).Splits

@[simp]

theorem Polynomial.Splits.zero {R : Type u_1} [Semiring R] : Splits 0

@[simp]

theorem Polynomial.Splits.one {R : Type u_1} [Semiring R] : Splits 1

@[simp]

theorem Polynomial.Splits.X_add_C {R : Type u_1} [Semiring R] (a : R) : (X + C a).Splits

@[simp]

theorem Polynomial.Splits.X {R : Type u_1} [Semiring R] : X.Splits

@[simp]

theorem Polynomial.Splits.mul {R : Type u_1} [Semiring R] {f g : Polynomial R} (hf : f.Splits) (hg : g.Splits) : (f * g).Splits

theorem Polynomial.Splits.C_mul {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.Splits) (a : R) : (C a * f).Splits

@[simp]

theorem Polynomial.Splits.listProd {R : Type u_1} [Semiring R] {l : List (Polynomial R)} (h : ∀ f ∈ l, f.Splits) : l.prod.Splits

@[simp]

theorem Polynomial.Splits.pow {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.Splits) (n : ℕ) : (f ^ n).Splits

theorem Polynomial.Splits.X_pow {R : Type u_1} [Semiring R] (n : ℕ) : (X ^ n).Splits

theorem Polynomial.Splits.C_mul_X_pow {R : Type u_1} [Semiring R] (a : R) (n : ℕ) : (C a * X ^ n).Splits

@[simp]

theorem Polynomial.Splits.monomial {R : Type u_1} [Semiring R] (n : ℕ) (a : R) : ((Polynomial.monomial n) a).Splits

theorem Polynomial.Splits.map {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.Splits) {S : Type u_2} [Semiring S] (i : R →+* S) : (map i f).Splits

theorem Polynomial.splits_of_natDegree_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.natDegree = 0) : f.Splits

theorem Polynomial.splits_of_degree_le_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.degree ≤ 0) : f.Splits

theorem IsUnit.splits {R : Type u_1} [Semiring R] [NoZeroDivisors R] {f : Polynomial R} (hf : IsUnit f) : f.Splits

@[deprecated IsUnit.splits (since := "2025-11-27")]

theorem Polynomial.splits_of_isUnit {R : Type u_1} [Semiring R] [NoZeroDivisors R] {f : Polynomial R} (hf : IsUnit f) : f.Splits

Alias of IsUnit.splits.

theorem Polynomial.splits_of_natDegree_le_one_of_invertible {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.natDegree ≤ 1) (h : Invertible f.leadingCoeff) : f.Splits

theorem Polynomial.splits_of_degree_le_one_of_invertible {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.degree ≤ 1) (h : Invertible f.leadingCoeff) : f.Splits

theorem Polynomial.splits_of_natDegree_le_one_of_monic {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.natDegree ≤ 1) (h : f.Monic) : f.Splits

theorem Polynomial.splits_of_degree_le_one_of_monic {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f.degree ≤ 1) (h : f.Monic) : f.Splits

@[simp]

theorem Polynomial.Splits.multisetProd {R : Type u_1} [CommSemiring R] {m : Multiset (Polynomial R)} (hm : ∀ f ∈ m, f.Splits) : m.prod.Splits

@[simp]

theorem Polynomial.Splits.prod {R : Type u_1} [CommSemiring R] {ι : Type u_2} {f : ι → Polynomial R} {s : Finset ι} (h : ∀ i ∈ s, (f i).Splits) : (∏ i ∈ s, f i).Splits

theorem Polynomial.Splits.taylor {R : Type u_1} [CommSemiring R] {p : Polynomial R} (hp : p.Splits) (r : R) : ((Polynomial.taylor r) p).Splits

theorem Polynomial.splits_iff_exists_multiset' {R : Type u_1} [CommSemiring R] {f : Polynomial R} : f.Splits ↔ ∃ (m : Multiset R), f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X + C x) m).prod

See splits_iff_exists_multiset for the version with subtraction.

theorem Polynomial.Splits.natDegree_le_one_of_irreducible {R : Type u_1} [CommSemiring R] {f : Polynomial R} (hf : f.Splits) (h : Irreducible f) : f.natDegree ≤ 1

theorem Polynomial.Splits.degree_le_one_of_irreducible {R : Type u_1} [CommSemiring R] {f : Polynomial R} (hf : f.Splits) (h : Irreducible f) : f.degree ≤ 1

theorem Polynomial.Splits.comp_of_natDegree_le_one_of_invertible {R : Type u_1} [CommSemiring R] {f g : Polynomial R} (hf : f.Splits) (hg : g.natDegree ≤ 1) (h : Invertible g.leadingCoeff) : (f.comp g).Splits

theorem Polynomial.Splits.comp_of_degree_le_one_of_invertible {R : Type u_1} [CommSemiring R] {f g : Polynomial R} (hf : f.Splits) (hg : g.degree ≤ 1) (h : Invertible g.leadingCoeff) : (f.comp g).Splits

theorem Polynomial.Splits.comp_of_natDegree_le_one_of_monic {R : Type u_1} [CommSemiring R] {f g : Polynomial R} (hf : f.Splits) (hg : g.natDegree ≤ 1) (h : g.Monic) : (f.comp g).Splits

theorem Polynomial.Splits.comp_of_degree_le_one_of_monic {R : Type u_1} [CommSemiring R] {f g : Polynomial R} (hf : f.Splits) (hg : g.degree ≤ 1) (h : g.Monic) : (f.comp g).Splits

theorem Polynomial.Splits.comp_X_add_C {R : Type u_1} [CommSemiring R] {f : Polynomial R} (hf : f.Splits) (a : R) : (f.comp (X + C a)).Splits

@[simp]

theorem Polynomial.Splits.X_sub_C {R : Type u_1} [Ring R] (a : R) : (X - C a).Splits

theorem Polynomial.Splits.neg {R : Type u_1} [Ring R] {f : Polynomial R} (hf : f.Splits) : (-f).Splits

@[simp]

theorem Polynomial.splits_neg_iff {R : Type u_1} [Ring R] {f : Polynomial R} : (-f).Splits ↔ f.Splits

theorem Polynomial.Splits.comp_neg_X {R : Type u_1} [Ring R] {f : Polynomial R} (hf : f.Splits) : (f.comp (-X)).Splits

theorem Polynomial.splits_iff_exists_multiset {R : Type u_1} [CommRing R] {f : Polynomial R} : f.Splits ↔ ∃ (m : Multiset R), f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X - C x) m).prod

theorem Polynomial.Splits.exists_eval_eq_zero {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hf0 : f.degree ≠ 0) : ∃ (a : R), eval a f = 0

noncomputable def Polynomial.rootOfSplits {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hfd : f.degree ≠ 0) : R

Pick a root of a polynomial that splits.

Equations

• Polynomial.rootOfSplits hf hfd = Classical.choose ⋯

@[simp]

theorem Polynomial.eval_rootOfSplits {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (hfd : f.degree ≠ 0) : eval (rootOfSplits hf hfd) f = 0

theorem Polynomial.Splits.comp_X_sub_C {R : Type u_1} [CommRing R] {f : Polynomial R} (hf : f.Splits) (a : R) : (f.comp (X - C a)).Splits

theorem Polynomial.Splits.eq_prod_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) : f = C f.leadingCoeff * (Multiset.map (fun (x : R) => X - C x) f.roots).prod

theorem Polynomial.Splits.eq_prod_roots_of_monic {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) : f = (Multiset.map (fun (x : R) => X - C x) f.roots).prod

theorem Polynomial.Splits.eval_eq_prod_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (x : R) : eval x f = f.leadingCoeff * (Multiset.map (fun (x_1 : R) => x - x_1) f.roots).prod

theorem Polynomial.Splits.eval_eq_prod_roots_of_monic {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) (x : R) : eval x f = (Multiset.map (fun (x_1 : R) => x - x_1) f.roots).prod

theorem Polynomial.Splits.aeval_eq_prod_aroots {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} [CommRing A] [IsDomain A] [IsSimpleRing R] [Algebra R A] (hf : (map (algebraMap R A) f).Splits) (x : A) : (aeval x) f = (algebraMap R A) f.leadingCoeff * (Multiset.map (fun (x_1 : A) => x - x_1) (f.aroots A)).prod

theorem Polynomial.Splits.aeval_eq_prod_aroots_of_monic {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} [CommRing A] [IsDomain A] [IsSimpleRing R] [Algebra R A] (hf : (map (algebraMap R A) f).Splits) (hm : f.Monic) (x : A) : (aeval x) f = (Multiset.map (fun (x_1 : A) => x - x_1) (f.aroots A)).prod

theorem Polynomial.Splits.eval_derivative {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] [DecidableEq R] (hf : f.Splits) (x : R) : eval x (derivative f) = f.leadingCoeff * (Multiset.map (fun (a : R) => (Multiset.map (fun (x_1 : R) => x - x_1) (f.roots.erase a)).prod) f.roots).sum

theorem Polynomial.Splits.eval_root_derivative {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] [DecidableEq R] (hf : f.Splits) (hm : f.Monic) {x : R} (hx : x ∈ f.roots) : eval x (derivative f) = (Multiset.map (fun (x_1 : R) => x - x_1) (f.roots.erase x)).prod

Let f be a monic polynomial over that splits. Let x be a root of f. Then $f'(r) = \prod_{a}(x-a)$, where the product in the RHS is taken over all roots of f, with the multiplicity of x reduced by one.

theorem Polynomial.Splits.of_splits_map {R : Type u_1} [CommRing R] {f : Polynomial R} {S : Type u_2} [CommRing S] [IsDomain S] [IsSimpleRing R] (i : R →+* S) (hf : (map i f).Splits) (hi : ∀ a ∈ (map i f).roots, a ∈ i.range) : f.Splits

theorem Polynomial.Splits.eq_X_sub_C_of_single_root {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) {x : R} (hr : f.roots = {x}) : f = C f.leadingCoeff * (X - C x)

theorem Polynomial.Splits.natDegree_eq_card_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) : f.natDegree = f.roots.card

theorem Polynomial.Splits.degree_eq_card_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hf0 : f ≠ 0) : f.degree = ↑f.roots.card

theorem Polynomial.splits_iff_card_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] : f.Splits ↔ f.roots.card = f.natDegree

A polynomial splits if and only if it has as many roots as its degree.

theorem Polynomial.Splits.roots_ne_zero {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hf0 : f.natDegree ≠ 0) : f.roots ≠ 0

theorem Polynomial.Splits.map_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] {S : Type u_2} [CommRing S] [IsDomain S] [IsSimpleRing R] (hf : f.Splits) (i : R →+* S) : (map i f).roots = Multiset.map (⇑i) f.roots

theorem Polynomial.Splits.mem_range_of_isRoot {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] {S : Type u_2} [CommRing S] [IsDomain S] [IsSimpleRing R] (hf : f.Splits) (hf0 : f ≠ 0) {i : R →+* S} {x : S} (hx : (map i f).IsRoot x) : x ∈ i.range

theorem Polynomial.Splits.image_rootSet {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} {B : Type u_3} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] [IsSimpleRing A] [Algebra R A] [Algebra R B] (hf : (map (algebraMap R A) f).Splits) (g : A →ₐ[R] B) : ⇑g '' f.rootSet A = f.rootSet B

theorem Polynomial.Splits.adjoin_rootSet_eq_range {R : Type u_1} [CommRing R] {f : Polynomial R} {A : Type u_2} {B : Type u_3} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] [IsSimpleRing A] [Algebra R A] [Algebra R B] (hf : (map (algebraMap R A) f).Splits) (g : A →ₐ[R] B) : Algebra.adjoin R (f.rootSet B) = g.range ↔ Algebra.adjoin R (f.rootSet A) = ⊤

theorem Polynomial.Splits.coeff_zero_eq_leadingCoeff_mul_prod_roots {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) : f.coeff 0 = (-1) ^ f.natDegree * f.leadingCoeff * f.roots.prod

theorem Polynomial.Splits.coeff_zero_eq_prod_roots_of_monic {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) : f.coeff 0 = (-1) ^ f.natDegree * f.roots.prod

If f is a monic polynomial that splits, then coeff f 0 equals the product of the roots.

theorem Polynomial.Splits.nextCoeff_eq_neg_sum_roots_mul_leadingCoeff {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) : f.nextCoeff = -f.leadingCoeff * f.roots.sum

theorem Polynomial.Splits.nextCoeff_eq_neg_sum_roots_of_monic {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) : f.nextCoeff = -f.roots.sum

If f is a monic polynomial that splits, then f.nextCoeff equals the negative of the sum of the roots.

theorem Polynomial.splits_X_sub_C_mul_iff {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] {a : R} : ((X - C a) * f).Splits ↔ f.Splits

theorem Polynomial.splits_mul_iff {R : Type u_1} [CommRing R] {f g : Polynomial R} [IsDomain R] (hf₀ : f ≠ 0) (hg₀ : g ≠ 0) : (f * g).Splits ↔ f.Splits ∧ g.Splits

theorem Polynomial.Splits.splits_of_dvd {R : Type u_1} [CommRing R] {f g : Polynomial R} [IsDomain R] (hg : g.Splits) (hg₀ : g ≠ 0) (hfg : f ∣ g) : f.Splits

@[deprecated Polynomial.Splits.splits_of_dvd (since := "2025-11-27")]

theorem Polynomial.Splits.of_dvd {R : Type u_1} [CommRing R] {f g : Polynomial R} [IsDomain R] (hg : g.Splits) (hg₀ : g ≠ 0) (hfg : f ∣ g) : f.Splits

Alias of Polynomial.Splits.splits_of_dvd.

theorem Polynomial.splits_prod_iff {R : Type u_1} [CommRing R] [IsDomain R] {ι : Type u_2} {f : ι → Polynomial R} {s : Finset ι} (hf : ∀ i ∈ s, f i ≠ 0) : (∏ x ∈ s, f x).Splits ↔ ∀ x ∈ s, (f x).Splits

theorem Polynomial.Splits.splits {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) : f = 0 ∨ ∀ {g : Polynomial R}, Irreducible g → g ∣ f → g.degree ≤ 1

theorem Polynomial.map_sub_sprod_roots_eq_prod_map_eval {R : Type u_1} [CommRing R] [IsDomain R] (s : Multiset R) (g : Polynomial R) (hg : g.Monic) (hg' : g.Splits) : (Multiset.map (fun (ij : R × R) => ij.1 - ij.2) (s ×ˢ g.roots)).prod = (Multiset.map (fun (x : R) => eval x g) s).prod

theorem Polynomial.map_sub_roots_sprod_eq_prod_map_eval {R : Type u_1} [CommRing R] [IsDomain R] (s : Multiset R) (g : Polynomial R) (hg : g.Monic) (hg' : g.Splits) : (Multiset.map (fun (ij : R × R) => ij.1 - ij.2) (g.roots ×ˢ s)).prod = (-1) ^ (s.card * g.roots.card) * (Multiset.map (fun (x : R) => eval x g) s).prod

theorem Polynomial.Splits.of_natDegree_le_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.natDegree ≤ 1) : f.Splits

theorem Polynomial.Splits.of_natDegree_eq_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.natDegree = 1) : f.Splits

theorem Polynomial.Splits.of_degree_le_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.degree ≤ 1) : f.Splits

theorem Polynomial.Splits.of_degree_eq_one {R : Type u_1} [DivisionSemiring R] {f : Polynomial R} (hf : f.degree = 1) : f.Splits

theorem Polynomial.Splits.dvd_of_roots_le_roots {R : Type u_1} [Field R] {f g : Polynomial R} (hp : f.Splits) (hp0 : f ≠ 0) (hq : f.roots ≤ g.roots) : f ∣ g

theorem Polynomial.Splits.dvd_iff_roots_le_roots {R : Type u_1} [Field R] {f g : Polynomial R} (hf : f.Splits) (hf0 : f ≠ 0) (hg0 : g ≠ 0) : f ∣ g ↔ f.roots ≤ g.roots

theorem Polynomial.Splits.comp_of_natDegree_le_one {R : Type u_1} [Field R] {f g : Polynomial R} (hf : f.Splits) (hg : g.natDegree ≤ 1) : (f.comp g).Splits

theorem Polynomial.Splits.comp_of_degree_le_one {R : Type u_1} [Field R] {f g : Polynomial R} (hf : f.Splits) (hg : g.degree ≤ 1) : (f.comp g).Splits

theorem Polynomial.splits_iff_comp_splits_of_natDegree_eq_one {R : Type u_1} [Field R] {f g : Polynomial R} (hg : g.natDegree = 1) : f.Splits ↔ (f.comp g).Splits

theorem Polynomial.splits_iff_comp_splits_of_degree_eq_one {R : Type u_1} [Field R] {f g : Polynomial R} (hg : g.degree = 1) : f.Splits ↔ (f.comp g).Splits

theorem Polynomial.Splits.degree_eq_one_of_irreducible {R : Type u_1} [Field R] {f : Polynomial R} (hf : f.Splits) (h : Irreducible f) : f.degree = 1

theorem Polynomial.Splits.natDegree_eq_one_of_irreducible {R : Type u_1} [Field R] {f : Polynomial R} (hf : f.Splits) (h : Irreducible f) : f.natDegree = 1

theorem Polynomial.Splits.eval_derivative_eq_eval_mul_sum {R : Type u_1} [Field R] {f : Polynomial R} (hf : f.Splits) {x : R} (hx : eval x f ≠ 0) : eval x (derivative f) = eval x f * (Multiset.map (fun (z : R) => 1 / (x - z)) f.roots).sum

theorem Polynomial.Splits.eval_derivative_div_eval_of_ne_zero {R : Type u_1} [Field R] {f : Polynomial R} (hf : f.Splits) {x : R} (hx : eval x f ≠ 0) : eval x (derivative f) / eval x f = (Multiset.map (fun (z : R) => 1 / (x - z)) f.roots).sum

theorem Polynomial.Splits.mem_subfield_of_isRoot {R : Type u_1} [Field R] (F : Subfield R) {f : Polynomial ↥F} (hf : f.Splits) (hf0 : f ≠ 0) {x : R} (hx : (map F.subtype f).IsRoot x) : x ∈ F

theorem Polynomial.splits_iff_splits {R : Type u_1} [Field R] {f : Polynomial R} : f.Splits ↔ f = 0 ∨ ∀ {g : Polynomial R}, Irreducible g → g ∣ f → g.degree = 1