Split polynomials

A polynomial f : K[X] splits over a field extension L of K if it is zero or all of its irreducible factors over L have degree 1.

Main definitions

• Polynomial.Splits i f: A predicate on a homomorphism i : K →+* L from a commutative ring to a field and a polynomial f saying that f.map i is zero or all of its irreducible factors over L have degree 1.

def Polynomial.Splits {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) (f : Polynomial K) : Prop

A polynomial Splits iff it is zero or all of its irreducible factors have degree 1.

@[simp]

theorem Polynomial.splits_zero {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) : Splits i 0

theorem Polynomial.splits_of_map_eq_C {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} {a : L} (h : map i f = C a) : Splits i f

@[simp]

theorem Polynomial.splits_C {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) (a : K) : Splits i (C a)

theorem Polynomial.splits_of_map_degree_eq_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : (map i f).degree = 1) : Splits i f

theorem Polynomial.splits_of_degree_le_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : f.degree ≤ 1) : Splits i f

theorem Polynomial.splits_of_degree_eq_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : f.degree = 1) : Splits i f

theorem Polynomial.splits_of_natDegree_le_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : f.natDegree ≤ 1) : Splits i f

theorem Polynomial.splits_of_natDegree_eq_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : f.natDegree = 1) : Splits i f

theorem Polynomial.splits_mul {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f g : Polynomial K} (hf : Splits i f) (hg : Splits i g) : Splits i (f * g)

theorem Polynomial.splits_of_splits_mul' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f g : Polynomial K} (hfg : map i (f * g) ≠ 0) (h : Splits i (f * g)) : Splits i f ∧ Splits i g

theorem Polynomial.splits_map_iff {F : Type u} {K : Type v} {L : Type w} [CommRing K] [Field L] [Field F] (i : K →+* L) (j : L →+* F) {f : Polynomial K} : Splits j (map i f) ↔ Splits (j.comp i) f

theorem Polynomial.splits_one {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) : Splits i 1

theorem Polynomial.splits_of_isUnit {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) [IsDomain K] {u : Polynomial K} (hu : IsUnit u) : Splits i u

theorem Polynomial.splits_X_sub_C {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {x : K} : Splits i (X - C x)

theorem Polynomial.splits_X {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) : Splits i X

theorem Polynomial.splits_prod {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {ι : Type u} {s : ι → Polynomial K} {t : Finset ι} : (∀ j ∈ t, Splits i (s j)) → Splits i (∏ x ∈ t, s x)

theorem Polynomial.splits_pow {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Splits i f) (n : ℕ) : Splits i (f ^ n)

theorem Polynomial.splits_X_pow {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) (n : ℕ) : Splits i (X ^ n)

theorem Polynomial.splits_id_iff_splits {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} : Splits (RingHom.id L) (map i f) ↔ Splits i f

theorem Polynomial.Splits.comp_of_map_degree_le_one {K : Type v} {L : Type w} [CommRing K] [Field L] {i : K →+* L} {f p : Polynomial K} (hd : (map i p).degree ≤ 1) (h : Splits i f) : Splits i (f.comp p)

theorem Polynomial.splits_iff_comp_splits_of_degree_eq_one {K : Type v} {L : Type w} [CommRing K] [Field L] {i : K →+* L} {f p : Polynomial K} (hd : (map i p).degree = 1) : Splits i f ↔ Splits i (f.comp p)

theorem Polynomial.Splits.comp_of_degree_le_one {K : Type v} {L : Type w} [CommRing K] [Field L] {i : K →+* L} {f p : Polynomial K} (hd : p.degree ≤ 1) (h : Splits i f) : Splits i (f.comp p)

This is a weaker variant of Splits.comp_of_map_degree_le_one, but its conditions are easier to check.

theorem Polynomial.Splits.comp_X_sub_C {K : Type v} {L : Type w} [CommRing K] [Field L] {i : K →+* L} (a : K) {f : Polynomial K} (h : Splits i f) : Splits i (f.comp (X - C a))

theorem Polynomial.Splits.comp_X_add_C {K : Type v} {L : Type w} [CommRing K] [Field L] {i : K →+* L} (a : K) {f : Polynomial K} (h : Splits i f) : Splits i (f.comp (X + C a))

theorem Polynomial.Splits.comp_neg_X {K : Type v} {L : Type w} [CommRing K] [Field L] {i : K →+* L} {f : Polynomial K} (h : Splits i f) : Splits i (f.comp (-X))

theorem Polynomial.exists_root_of_splits' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hs : Splits i f) (hf0 : (map i f).degree ≠ 0) : ∃ (x : L), eval₂ i x f = 0

theorem Polynomial.roots_ne_zero_of_splits' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hs : Splits i f) (hf0 : (map i f).natDegree ≠ 0) : (map i f).roots ≠ 0

def Polynomial.rootOfSplits' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Splits i f) (hfd : (map i f).degree ≠ 0) : L

Pick a root of a polynomial that splits. See rootOfSplits for polynomials over a field which has simpler assumptions.

theorem Polynomial.map_rootOfSplits' {K : Type v} {L : Type w} [CommRing K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Splits i f) (hfd : (map i f).degree ≠ 0) : eval₂ i (rootOfSplits' i hf hfd) f = 0

theorem Polynomial.natDegree_eq_card_roots' {K : Type v} {L : Type w} [CommRing K] [Field L] {p : Polynomial K} {i : K →+* L} (hsplit : Splits i p) : (map i p).natDegree = (map i p).roots.card

theorem Polynomial.degree_eq_card_roots' {K : Type v} {L : Type w} [CommRing K] [Field L] {p : Polynomial K} {i : K →+* L} (p_ne_zero : map i p ≠ 0) (hsplit : Splits i p) : (map i p).degree = ↑(map i p).roots.card

theorem Polynomial.aeval_root_of_mapAlg_eq_multiset_prod_X_sub_C {R : Type u_1} {L : Type w} [CommSemiring R] [CommRing L] [Algebra R L] (s : Multiset L) {x : L} (hx : x ∈ s) {p : Polynomial R} (hp : (mapAlg R L) p = (Multiset.map (fun (a : L) => X - C a) s).prod) : (aeval x) p = 0

theorem Polynomial.splits_iff {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) (f : Polynomial K) : Splits i f ↔ f = 0 ∨ ∀ {g : Polynomial L}, Irreducible g → g ∣ map i f → g.degree = 1

This lemma is for polynomials over a field.

theorem Polynomial.Splits.def {K : Type v} {L : Type w} [Field K] [Field L] {i : K →+* L} {f : Polynomial K} (h : Splits i f) : f = 0 ∨ ∀ {g : Polynomial L}, Irreducible g → g ∣ map i f → g.degree = 1

This lemma is for polynomials over a field.

theorem Polynomial.splits_of_splits_mul {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f g : Polynomial K} (hfg : f * g ≠ 0) (h : Splits i (f * g)) : Splits i f ∧ Splits i g

theorem Polynomial.splits_of_splits_of_dvd {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f g : Polynomial K} (hf0 : f ≠ 0) (hf : Splits i f) (hgf : g ∣ f) : Splits i g

theorem Polynomial.splits_of_splits_gcd_left {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) [DecidableEq K] {f g : Polynomial K} (hf0 : f ≠ 0) (hf : Splits i f) : Splits i (EuclideanDomain.gcd f g)

theorem Polynomial.splits_of_splits_gcd_right {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) [DecidableEq K] {f g : Polynomial K} (hg0 : g ≠ 0) (hg : Splits i g) : Splits i (EuclideanDomain.gcd f g)

theorem Polynomial.splits_mul_iff {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f g : Polynomial K} (hf : f ≠ 0) (hg : g ≠ 0) : Splits i (f * g) ↔ Splits i f ∧ Splits i g

theorem Polynomial.splits_prod_iff {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {ι : Type u} {s : ι → Polynomial K} {t : Finset ι} : (∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j))

theorem Polynomial.degree_eq_one_of_irreducible_of_splits {K : Type v} [Field K] {p : Polynomial K} (hp : Irreducible p) (hp_splits : Splits (RingHom.id K) p) : p.degree = 1

theorem Polynomial.exists_root_of_splits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hs : Splits i f) (hf0 : f.degree ≠ 0) : ∃ (x : L), eval₂ i x f = 0

theorem Polynomial.roots_ne_zero_of_splits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hs : Splits i f) (hf0 : f.natDegree ≠ 0) : (map i f).roots ≠ 0

def Polynomial.rootOfSplits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Splits i f) (hfd : f.degree ≠ 0) : L

Pick a root of a polynomial that splits. This version is for polynomials over a field and has simpler assumptions.

theorem Polynomial.rootOfSplits'_eq_rootOfSplits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Splits i f) (hfd : (map i f).degree ≠ 0) : rootOfSplits' i hf hfd = rootOfSplits i hf ⋯

rootOfSplits' is definitionally equal to rootOfSplits.

theorem Polynomial.map_rootOfSplits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Splits i f) (hfd : f.degree ≠ 0) : eval₂ i (rootOfSplits i hf hfd) f = 0

theorem Polynomial.natDegree_eq_card_roots {K : Type v} {L : Type w} [Field K] [Field L] {p : Polynomial K} {i : K →+* L} (hsplit : Splits i p) : p.natDegree = (map i p).roots.card

theorem Polynomial.degree_eq_card_roots {K : Type v} {L : Type w} [Field K] [Field L] {p : Polynomial K} {i : K →+* L} (p_ne_zero : p ≠ 0) (hsplit : Splits i p) : p.degree = ↑(map i p).roots.card

theorem Polynomial.roots_map {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (hf : Splits (RingHom.id K) f) : (map i f).roots = Multiset.map (⇑i) f.roots

theorem Polynomial.Splits.mem_subfield_of_isRoot {K : Type v} [Field K] (F : Subfield K) {f : Polynomial ↥F} (hnz : f ≠ 0) (hf : Splits (RingHom.id ↥F) f) {x : K} (hx : (map F.subtype f).IsRoot x) : x ∈ F

theorem Polynomial.image_rootSet {R : Type u_1} {K : Type v} {L : Type w} [CommRing R] [Field K] [Field L] [Algebra R K] [Algebra R L] {p : Polynomial R} (h : Splits (algebraMap R K) p) (f : K →ₐ[R] L) : ⇑f '' p.rootSet K = p.rootSet L

theorem Polynomial.adjoin_rootSet_eq_range {R : Type u_1} {K : Type v} {L : Type w} [CommRing R] [Field K] [Field L] [Algebra R K] [Algebra R L] {p : Polynomial R} (h : Splits (algebraMap R K) p) (f : K →ₐ[R] L) : Algebra.adjoin R (p.rootSet L) = f.range ↔ Algebra.adjoin R (p.rootSet K) = ⊤

theorem Polynomial.eq_prod_roots_of_splits {K : Type v} {L : Type w} [Field K] [Field L] {p : Polynomial K} {i : K →+* L} (hsplit : Splits i p) : map i p = C (i p.leadingCoeff) * (Multiset.map (fun (a : L) => X - C a) (map i p).roots).prod

theorem Polynomial.eq_prod_roots_of_splits_id {K : Type v} [Field K] {p : Polynomial K} (hsplit : Splits (RingHom.id K) p) : p = C p.leadingCoeff * (Multiset.map (fun (a : K) => X - C a) p.roots).prod

theorem Polynomial.Splits.dvd_of_roots_le_roots {K : Type v} [Field K] {p q : Polynomial K} (hp : Splits (RingHom.id K) p) (hp0 : p ≠ 0) (hq : p.roots ≤ q.roots) : p ∣ q

theorem Polynomial.Splits.dvd_iff_roots_le_roots {K : Type v} [Field K] {p q : Polynomial K} (hp : Splits (RingHom.id K) p) (hp0 : p ≠ 0) (hq0 : q ≠ 0) : p ∣ q ↔ p.roots ≤ q.roots

theorem Polynomial.aeval_eq_prod_aroots_sub_of_splits {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] {p : Polynomial K} (hsplit : Splits (algebraMap K L) p) (v : L) : (aeval v) p = (algebraMap K L) p.leadingCoeff * (Multiset.map (fun (a : L) => v - a) (p.aroots L)).prod

theorem Polynomial.eval_eq_prod_roots_sub_of_splits_id {K : Type v} [Field K] {p : Polynomial K} (hsplit : Splits (RingHom.id K) p) (v : K) : eval v p = p.leadingCoeff * (Multiset.map (fun (a : K) => v - a) p.roots).prod

theorem Polynomial.eq_prod_roots_of_monic_of_splits_id {K : Type v} [Field K] {p : Polynomial K} (m : p.Monic) (hsplit : Splits (RingHom.id K) p) : p = (Multiset.map (fun (a : K) => X - C a) p.roots).prod

theorem Polynomial.aeval_eq_prod_aroots_sub_of_monic_of_splits {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] {p : Polynomial K} (m : p.Monic) (hsplit : Splits (algebraMap K L) p) (v : L) : (aeval v) p = (Multiset.map (fun (a : L) => v - a) (p.aroots L)).prod

theorem Polynomial.eval_eq_prod_roots_sub_of_monic_of_splits_id {K : Type v} [Field K] {p : Polynomial K} (m : p.Monic) (hsplit : Splits (RingHom.id K) p) (v : K) : eval v p = (Multiset.map (fun (a : K) => v - a) p.roots).prod

theorem Polynomial.eq_X_sub_C_of_splits_of_single_root {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {x : K} {h : Polynomial K} (h_splits : Splits i h) (h_roots : (map i h).roots = {i x}) : h = C h.leadingCoeff * (X - C x)

theorem Polynomial.mem_lift_of_splits_of_roots_mem_range (R : Type u_1) {K : Type v} [CommRing R] [Field K] [Algebra R K] {f : Polynomial K} (hs : Splits (RingHom.id K) f) (hm : f.Monic) (hr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range) : f ∈ lifts (algebraMap R K)

theorem Polynomial.splits_of_natDegree_eq_two {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} {x : L} (h₁ : f.natDegree = 2) (h₂ : eval₂ i x f = 0) : Splits i f

A polynomial of degree 2 with a root splits.

theorem Polynomial.splits_of_degree_eq_two {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} {x : L} (h₁ : f.degree = 2) (h₂ : eval₂ i x f = 0) : Splits i f

theorem Polynomial.splits_of_exists_multiset {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} {s : Multiset L} (hs : map i f = C (i f.leadingCoeff) * (Multiset.map (fun (a : L) => X - C a) s).prod) : Splits i f

theorem Polynomial.splits_of_splits_id {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} : Splits (RingHom.id K) f → Splits i f

theorem Polynomial.splits_iff_exists_multiset {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} : Splits i f ↔ ∃ (s : Multiset L), map i f = C (i f.leadingCoeff) * (Multiset.map (fun (a : L) => X - C a) s).prod

theorem Polynomial.splits_of_comp {F : Type u} {K : Type v} {L : Type w} [Field K] [Field L] [Field F] (i : K →+* L) (j : L →+* F) {f : Polynomial K} (h : Splits (j.comp i) f) (roots_mem_range : ∀ a ∈ (map (j.comp i) f).roots, a ∈ j.range) : Splits i f

theorem Polynomial.splits_id_of_splits {K : Type v} {L : Type w} [Field K] [Field L] (i : K →+* L) {f : Polynomial K} (h : Splits i f) (roots_mem_range : ∀ a ∈ (map i f).roots, a ∈ i.range) : Splits (RingHom.id K) f

theorem Polynomial.splits_comp_of_splits {R : Type u_1} {K : Type v} {L : Type w} [CommRing R] [Field K] [Field L] (i : R →+* K) (j : K →+* L) {f : Polynomial R} (h : Splits i f) : Splits (j.comp i) f

theorem Polynomial.splits_of_algHom {R : Type u_1} {K : Type v} {L : Type w} [CommRing R] [Field K] [Field L] [Algebra R K] [Algebra R L] {f : Polynomial R} (h : Splits (algebraMap R K) f) (e : K →ₐ[R] L) : Splits (algebraMap R L) f

theorem Polynomial.splits_of_isScalarTower {R : Type u_1} {K : Type v} (L : Type w) [CommRing R] [Field K] [Field L] [Algebra R K] [Algebra R L] {f : Polynomial R} [Algebra K L] [IsScalarTower R K L] (h : Splits (algebraMap R K) f) : Splits (algebraMap R L) f

theorem Polynomial.splits_iff_card_roots {K : Type v} [Field K] {p : Polynomial K} : Splits (RingHom.id K) p ↔ p.roots.card = p.natDegree

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

theorem Polynomial.aeval_root_derivative_of_splits {K : Type v} {L : Type w} [Field K] [Field L] [Algebra K L] [DecidableEq L] {P : Polynomial K} (hmo : P.Monic) (hP : Splits (algebraMap K L) P) {r : L} (hr : r ∈ P.aroots L) : (aeval r) (derivative P) = (Multiset.map (fun (a : L) => r - a) ((P.aroots L).erase r)).prod

theorem Polynomial.prod_roots_eq_coeff_zero_of_monic_of_splits {K : Type v} [Field K] {P : Polynomial K} (hmo : P.Monic) (hP : Splits (RingHom.id K) P) : P.coeff 0 = (-1) ^ P.natDegree * P.roots.prod

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

theorem Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split {K : Type v} [Field K] {P : Polynomial K} (hmo : P.Monic) (hP : Splits (RingHom.id K) P) : P.nextCoeff = -P.roots.sum

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