Separable polynomials

We define a polynomial to be separable if it is coprime with its derivative. We prove basic properties about separable polynomials here.

Main definitions

• Polynomial.Separable f: a polynomial f is separable iff it is coprime with its derivative.
• IsSeparable K x: an element x is separable over K iff the minimal polynomial of x over K is separable.
• Algebra.IsSeparable K L: L is separable over K iff every element in L is separable over K.

def Polynomial.Separable {R : Type u} [CommSemiring R] (f : Polynomial R) : Prop

A polynomial is separable iff it is coprime with its derivative.

Equations

• f.Separable = IsCoprime f (Polynomial.derivative f)

theorem Polynomial.separable_def {R : Type u} [CommSemiring R] (f : Polynomial R) : f.Separable ↔ IsCoprime f (derivative f)

theorem Polynomial.separable_def' {R : Type u} [CommSemiring R] (f : Polynomial R) : f.Separable ↔ ∃ (a : Polynomial R) (b : Polynomial R), a * f + b * derivative f = 1

theorem Polynomial.not_separable_zero {R : Type u} [CommSemiring R] [Nontrivial R] : ¬Separable 0

theorem Polynomial.Separable.ne_zero {R : Type u} [CommSemiring R] [Nontrivial R] {f : Polynomial R} (h : f.Separable) : f ≠ 0

@[simp]

theorem Polynomial.separable_one {R : Type u} [CommSemiring R] : Separable 1

theorem Polynomial.separable_of_subsingleton {R : Type u} [CommSemiring R] [Subsingleton R] (f : Polynomial R) : f.Separable

theorem Polynomial.separable_X_add_C {R : Type u} [CommSemiring R] (a : R) : (X + C a).Separable

theorem Polynomial.separable_X {R : Type u} [CommSemiring R] : X.Separable

theorem Polynomial.separable_C {R : Type u} [CommSemiring R] (r : R) : (C r).Separable ↔ IsUnit r

theorem Polynomial.Separable.of_mul_left {R : Type u} [CommSemiring R] {f g : Polynomial R} (h : (f * g).Separable) : f.Separable

theorem Polynomial.Separable.of_mul_right {R : Type u} [CommSemiring R] {f g : Polynomial R} (h : (f * g).Separable) : g.Separable

theorem Polynomial.Separable.of_dvd {R : Type u} [CommSemiring R] {f g : Polynomial R} (hf : f.Separable) (hfg : g ∣ f) : g.Separable

theorem Polynomial.separable_gcd_left {F : Type u_1} [Field F] [DecidableEq (Polynomial F)] {f : Polynomial F} (hf : f.Separable) (g : Polynomial F) : (EuclideanDomain.gcd f g).Separable

theorem Polynomial.separable_gcd_right {F : Type u_1} [Field F] [DecidableEq (Polynomial F)] {g : Polynomial F} (f : Polynomial F) (hg : g.Separable) : (EuclideanDomain.gcd f g).Separable

theorem Polynomial.Separable.isCoprime {R : Type u} [CommSemiring R] {f g : Polynomial R} (h : (f * g).Separable) : IsCoprime f g

theorem Polynomial.Separable.of_pow' {R : Type u} [CommSemiring R] {f : Polynomial R} {n : ℕ} (_h : (f ^ n).Separable) : IsUnit f ∨ f.Separable ∧ n = 1 ∨ n = 0

theorem Polynomial.Separable.of_pow {R : Type u} [CommSemiring R] {f : Polynomial R} (hf : ¬IsUnit f) {n : ℕ} (hn : n ≠ 0) (hfs : (f ^ n).Separable) : f.Separable ∧ n = 1

theorem Polynomial.Separable.map {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] {p : Polynomial R} (h : p.Separable) {f : R →+* S} : (Polynomial.map f p).Separable

theorem Associated.separable {R : Type u} [CommSemiring R] {f g : Polynomial R} (ha : Associated f g) (h : f.Separable) : g.Separable

theorem Associated.separable_iff {R : Type u} [CommSemiring R] {f g : Polynomial R} (ha : Associated f g) : f.Separable ↔ g.Separable

theorem Polynomial.Separable.mul_unit {R : Type u} [CommSemiring R] {f g : Polynomial R} (hf : f.Separable) (hg : IsUnit g) : (f * g).Separable

theorem Polynomial.Separable.unit_mul {R : Type u} [CommSemiring R] {f g : Polynomial R} (hf : IsUnit f) (hg : g.Separable) : (f * g).Separable

theorem Polynomial.Separable.eval₂_derivative_ne_zero {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] [Nontrivial S] (f : R →+* S) {p : Polynomial R} (h : p.Separable) {x : S} (hx : eval₂ f x p = 0) : eval₂ f x (derivative p) ≠ 0

theorem Polynomial.Separable.aeval_derivative_ne_zero {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] [Nontrivial S] [Algebra R S] {p : Polynomial R} (h : p.Separable) {x : S} (hx : (aeval x) p = 0) : (aeval x) (derivative p) ≠ 0

theorem Polynomial.isUnit_of_self_mul_dvd_separable {R : Type u} [CommSemiring R] {p q : Polynomial R} (hp : p.Separable) (hq : q * q ∣ p) : IsUnit q

theorem Polynomial.emultiplicity_le_one_of_separable {R : Type u} [CommSemiring R] {p q : Polynomial R} (hq : ¬IsUnit q) (hsep : p.Separable) : emultiplicity q p ≤ 1

theorem Polynomial.Separable.squarefree {R : Type u} [CommSemiring R] {p : Polynomial R} (hsep : p.Separable) : Squarefree p

A separable polynomial is square-free.

See PerfectField.separable_iff_squarefree for the converse when the coefficients are a perfect field.

theorem Polynomial.separable_X_sub_C {R : Type u} [CommRing R] {x : R} : (X - C x).Separable

theorem Polynomial.Separable.mul {R : Type u} [CommRing R] {f g : Polynomial R} (hf : f.Separable) (hg : g.Separable) (h : IsCoprime f g) : (f * g).Separable

theorem Polynomial.separable_prod' {R : Type u} [CommRing R] {ι : Type u_1} {f : ι → Polynomial R} {s : Finset ι} : (∀ x ∈ s, ∀ y ∈ s, x ≠ y → IsCoprime (f x) (f y)) → (∀ x ∈ s, (f x).Separable) → (∏ x ∈ s, f x).Separable

theorem Polynomial.separable_prod {R : Type u} [CommRing R] {ι : Type u_1} [Fintype ι] {f : ι → Polynomial R} (h1 : Pairwise (Function.onFun IsCoprime f)) (h2 : ∀ (x : ι), (f x).Separable) : (∏ x : ι, f x).Separable

theorem Polynomial.Separable.inj_of_prod_X_sub_C {R : Type u} [CommRing R] [Nontrivial R] {ι : Type u_1} {f : ι → R} {s : Finset ι} (hfs : (∏ i ∈ s, (X - C (f i))).Separable) {x y : ι} (hx : x ∈ s) (hy : y ∈ s) (hfxy : f x = f y) : x = y

theorem Polynomial.Separable.injective_of_prod_X_sub_C {R : Type u} [CommRing R] [Nontrivial R] {ι : Type u_1} [Fintype ι] {f : ι → R} (hfs : (∏ i : ι, (X - C (f i))).Separable) : Function.Injective f

theorem Polynomial.nodup_of_separable_prod {R : Type u} [CommRing R] [Nontrivial R] {s : Multiset R} (hs : (Multiset.map (fun (a : R) => X - C a) s).prod.Separable) : s.Nodup

theorem Polynomial.separable_X_pow_sub_C_unit {R : Type u} [CommRing R] {n : ℕ} (u : Rˣ) (hn : IsUnit ↑n) : (X ^ n - C ↑u).Separable

If IsUnit n in a CommRing R, then X ^ n - u is separable for any unit u.

theorem Polynomial.separable_C_mul_X_pow_add_C_mul_X_add_C {R : Type u} [CommRing R] {n : ℕ} (a b c : R) (hn : ↑n = 0) (hb : IsUnit b) : (C a * X ^ n + C b * X + C c).Separable

If n = 0 in R and b is a unit, then a * X ^ n + b * X + c is separable.

theorem Polynomial.separable_C_mul_X_pow_add_C_mul_X_add_C' {R : Type u} [CommRing R] (p n : ℕ) (a b c : R) [CharP R p] (hn : p ∣ n) (hb : IsUnit b) : (C a * X ^ n + C b * X + C c).Separable

If R is of characteristic p, p ∣ n and b is a unit, then a * X ^ n + b * X + c is separable.

theorem Polynomial.rootMultiplicity_le_one_of_separable {R : Type u} [CommRing R] [Nontrivial R] {p : Polynomial R} (hsep : p.Separable) (x : R) : rootMultiplicity x p ≤ 1

theorem Polynomial.count_roots_le_one {R : Type u} [CommRing R] [IsDomain R] [DecidableEq R] {p : Polynomial R} (hsep : p.Separable) (x : R) : Multiset.count x p.roots ≤ 1

theorem Polynomial.nodup_roots {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hsep : p.Separable) : p.roots.Nodup

theorem Polynomial.separable_iff_derivative_ne_zero {F : Type u} [Field F] {f : Polynomial F} (hf : Irreducible f) : f.Separable ↔ derivative f ≠ 0

theorem Polynomial.separable_map {F : Type u} [Field F] {S : Type u_1} [CommRing S] [Nontrivial S] (f : F →+* S) {p : Polynomial F} : (map f p).Separable ↔ p.Separable

theorem Polynomial.separable_prod_X_sub_C_iff' {F : Type u} [Field F] {ι : Type u_1} {f : ι → F} {s : Finset ι} : (∏ i ∈ s, (X - C (f i))).Separable ↔ ∀ x ∈ s, ∀ y ∈ s, f x = f y → x = y

theorem Polynomial.separable_prod_X_sub_C_iff {F : Type u} [Field F] {ι : Type u_1} [Fintype ι] {f : ι → F} : (∏ i : ι, (X - C (f i))).Separable ↔ Function.Injective f

theorem Polynomial.separable_or {F : Type u} [Field F] (p : ℕ) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) : f.Separable ∨ ¬f.Separable ∧ ∃ (g : Polynomial F), Irreducible g ∧ (expand F p) g = f

theorem Polynomial.exists_separable_of_irreducible {F : Type u} [Field F] (p : ℕ) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) (hp : p ≠ 0) : ∃ (n : ℕ) (g : Polynomial F), g.Separable ∧ (expand F (p ^ n)) g = f

theorem Polynomial.isUnit_or_eq_zero_of_separable_expand {F : Type u} [Field F] (p : ℕ) [HF : CharP F p] {f : Polynomial F} (n : ℕ) (hp : 0 < p) (hf : ((expand F (p ^ n)) f).Separable) : IsUnit f ∨ n = 0

theorem Polynomial.unique_separable_of_irreducible {F : Type u} [Field F] (p : ℕ) [HF : CharP F p] {f : Polynomial F} (hf : Irreducible f) (hp : 0 < p) (n₁ : ℕ) (g₁ : Polynomial F) (hg₁ : g₁.Separable) (hgf₁ : (expand F (p ^ n₁)) g₁ = f) (n₂ : ℕ) (g₂ : Polynomial F) (hg₂ : g₂.Separable) (hgf₂ : (expand F (p ^ n₂)) g₂ = f) : n₁ = n₂ ∧ g₁ = g₂

theorem Polynomial.separable_X_pow_sub_C {F : Type u} [Field F] {n : ℕ} (a : F) (hn : ↑n ≠ 0) (ha : a ≠ 0) : (X ^ n - C a).Separable

If n ≠ 0 in F, then X ^ n - a is separable for any a ≠ 0.

theorem Polynomial.separable_X_pow_sub_C' {F : Type u} [Field F] (p n : ℕ) (a : F) [CharP F p] (hn : ¬p ∣ n) (ha : a ≠ 0) : (X ^ n - C a).Separable

If F is of characteristic p and p ∤ n, then X ^ n - a is separable for any a ≠ 0.

theorem Polynomial.X_pow_sub_one_separable_iff {F : Type u} [Field F] {n : ℕ} : (X ^ n - 1).Separable ↔ ↑n ≠ 0

In a field F, X ^ n - 1 is separable iff ↑n ≠ 0.

theorem Polynomial.card_rootSet_eq_natDegree {F : Type u} [Field F] {K : Type v} [Field K] [Algebra F K] {p : Polynomial F} (hsep : p.Separable) (hsplit : Splits (algebraMap F K) p) : Fintype.card ↑(p.rootSet K) = p.natDegree

theorem Polynomial.nodup_roots_iff_of_splits {F : Type u} [Field F] {f : Polynomial F} (hf : f ≠ 0) (h : Splits (RingHom.id F) f) : f.roots.Nodup ↔ f.Separable

If a non-zero polynomial splits, then it has no repeated roots on that field if and only if it is separable.

theorem Polynomial.nodup_aroots_iff_of_splits {F : Type u} [Field F] {K : Type v} [Field K] [Algebra F K] {f : Polynomial F} (hf : f ≠ 0) (h : Splits (algebraMap F K) f) : (f.aroots K).Nodup ↔ f.Separable

If a non-zero polynomial over F splits in K, then it has no repeated roots on K if and only if it is separable.

theorem Polynomial.card_rootSet_eq_natDegree_iff_of_splits {F : Type u} [Field F] {K : Type v} [Field K] [Algebra F K] {f : Polynomial F} (hf : f ≠ 0) (h : Splits (algebraMap F K) f) : Fintype.card ↑(f.rootSet K) = f.natDegree ↔ f.Separable

theorem Polynomial.eq_X_sub_C_of_separable_of_root_eq {F : Type u} [Field F] {K : Type v} [Field K] {i : F →+* K} {x : F} {h : Polynomial F} (h_sep : h.Separable) (h_root : eval x h = 0) (h_splits : Splits i h) (h_roots : ∀ y ∈ (map i h).roots, y = i x) : h = C h.leadingCoeff * (X - C x)

theorem Polynomial.exists_finset_of_splits {F : Type u} [Field F] {K : Type v} [Field K] (i : F →+* K) {f : Polynomial F} (sep : f.Separable) (sp : Splits i f) : ∃ (s : Finset K), map i f = C (i f.leadingCoeff) * ∏ a ∈ s, (X - C a)

theorem Irreducible.separable {F : Type u} [Field F] [CharZero F] {f : Polynomial F} (hf : Irreducible f) : f.Separable

def IsSeparable (F : Type u_1) {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] (x : K) : Prop

An element x of an algebra K over a commutative ring F is said to be separable, if its minimal polynomial over K is separable. Note that the minimal polynomial of any element not integral over F is defined to be 0, which is not a separable polynomial.

Equations

• IsSeparable F x = (minpoly F x).Separable

class Algebra.IsSeparable (F : Type u_1) (K : Type u_3) [CommRing F] [Ring K] [Algebra F K] : Prop

Typeclass for separable field extension: K is a separable field extension of F iff the minimal polynomial of every x : K is separable. This implies that K/F is an algebraic extension, because the minimal polynomial of a non-integral element is 0, which is not separable.

We define this for general (commutative) rings and only assume F and K are fields if this is needed for a proof.

• isSeparable' (x : K) : IsSeparable F x

theorem Algebra.isSeparable_def (F : Type u_1) (K : Type u_3) [CommRing F] [Ring K] [Algebra F K] : Algebra.IsSeparable F K ↔ ∀ (x : K), IsSeparable F x

theorem Algebra.IsSeparable.isSeparable (F : Type u_1) {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] [Algebra.IsSeparable F K] (x : K) : IsSeparable F x

theorem IsSeparable.isIntegral {F : Type u_1} {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] {x : K} (h : IsSeparable F x) : IsIntegral F x

If the minimal polynomial of x : K over F is separable, then x is integral over F, because the minimal polynomial of a non-integral element is 0, which is not separable.

theorem Algebra.IsSeparable.isIntegral (F : Type u_1) {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] [Algebra.IsSeparable F K] (x : K) : IsIntegral F x

instance Algebra.IsSeparable.isAlgebraic (F : Type u_1) (K : Type u_3) [CommRing F] [Ring K] [Algebra F K] [Nontrivial F] [Algebra.IsSeparable F K] : Algebra.IsAlgebraic F K

theorem Algebra.isSeparable_iff {F : Type u_1} {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] : Algebra.IsSeparable F K ↔ ∀ (x : K), IsIntegral F x ∧ IsSeparable F x

theorem IsSeparable.map {F : Type u_1} {L : Type u_2} {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] [Ring L] [Algebra F L] {x : K} (f : K →ₐ[F] L) (hf : Function.Injective ⇑f) (H : IsSeparable F x) : IsSeparable F (f x)

theorem AlgEquiv.isSeparable_iff {F : Type u_1} {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] {E : Type u_4} [Ring E] [Algebra F E] (e : K ≃ₐ[F] E) {x : K} : IsSeparable F (e x) ↔ IsSeparable F x

Transfer IsSeparable across an AlgEquiv.

theorem AlgEquiv.Algebra.isSeparable {F : Type u_1} {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] {E : Type u_4} [Ring E] [Algebra F E] (e : K ≃ₐ[F] E) [Algebra.IsSeparable F K] : Algebra.IsSeparable F E

Transfer Algebra.IsSeparable across an AlgEquiv.

theorem AlgEquiv.Algebra.isSeparable_iff {F : Type u_1} {K : Type u_3} [CommRing F] [Ring K] [Algebra F K] {E : Type u_4} [Ring E] [Algebra F E] (e : K ≃ₐ[F] E) : Algebra.IsSeparable F K ↔ Algebra.IsSeparable F E

theorem IsSeparable.tower_top {F : Type u_1} (L : Type u_2) [CommRing F] {E : Type u_4} [Field L] [Ring E] [Algebra F L] [Algebra F E] [Algebra L E] [IsScalarTower F L E] {x : E} (h : IsSeparable F x) : IsSeparable L x

If E / L / F is a scalar tower and x : E is separable over F, then it's also separable over L.

theorem Algebra.isSeparable_tower_top_of_isSeparable (F : Type u_1) (L : Type u_2) [CommRing F] (E : Type u_4) [Field L] [Ring E] [Algebra F L] [Algebra F E] [Algebra L E] [IsScalarTower F L E] [Algebra.IsSeparable F E] : Algebra.IsSeparable L E

If E / K / F is an extension tower, E is separable over F, then it's also separable over K.

theorem isSeparable_algebraMap {F : Type u_1} [Field F] {K : Type u_2} [Ring K] [Algebra F K] (x : F) : IsSeparable F ((algebraMap F K) x)

instance Algebra.isSeparable_self (F : Type u_1) [Field F] : Algebra.IsSeparable F F

theorem IsSeparable.of_integral (F : Type u_1) [Field F] {K : Type u_2} [Ring K] [Algebra F K] [IsDomain K] [Algebra.IsIntegral F K] [CharZero F] (x : K) : IsSeparable F x

@[instance 100]

instance Algebra.IsSeparable.of_integral (F : Type u_1) [Field F] (K : Type u_2) [Ring K] [Algebra F K] [IsDomain K] [Algebra.IsIntegral F K] [CharZero F] : Algebra.IsSeparable F K

A integral field extension in characteristic 0 is separable.

theorem IsSeparable.tower_bot {F : Type u_1} [Field F] {K : Type u_2} {E : Type u_3} [Field K] [Ring E] [Algebra F K] [Algebra F E] [Algebra K E] [Nontrivial E] [IsScalarTower F K E] {x : K} (h : IsSeparable F ((algebraMap K E) x)) : IsSeparable F x

If E / K / F is a scalar tower and algebraMap K E x is separable over F, then x is also separable over F.

theorem Algebra.isSeparable_tower_bot_of_isSeparable (F : Type u_1) [Field F] (K : Type u_2) (E : Type u_3) [Field K] [Ring E] [Algebra F K] [Algebra F E] [Algebra K E] [Nontrivial E] [IsScalarTower F K E] [h : Algebra.IsSeparable F E] : Algebra.IsSeparable F K

theorem IsSeparable.of_algHom {F : Type u_1} [Field F] {E : Type u_3} {E' : Type u_4} [Field E] [Field E'] [Algebra F E] [Algebra F E'] (f : E →ₐ[F] E') {x : E} (h : IsSeparable F (f x)) : IsSeparable F x

theorem Algebra.IsSeparable.of_algHom (F : Type u_1) [Field F] {E : Type u_3} (E' : Type u_4) [Field E] [Field E'] [Algebra F E] [Algebra F E'] (f : E →ₐ[F] E') [Algebra.IsSeparable F E'] : Algebra.IsSeparable F E

instance IntermediateField.isSeparable_tower_bot (F : Type u_1) [Field F] {K : Type u_2} [Field K] [Algebra F K] (M : IntermediateField F K) [Algebra.IsSeparable F K] : Algebra.IsSeparable F ↥M

instance IntermediateField.isSeparable_tower_top (F : Type u_1) [Field F] {K : Type u_2} [Field K] [Algebra F K] (M : IntermediateField F K) [Algebra.IsSeparable F K] : Algebra.IsSeparable (↥M) K

theorem IsSeparable.of_equiv_equiv {A₁ : Type u_1} {B₁ : Type u_2} {A₂ : Type u_3} {B₂ : Type u_4} [Field A₁] [Ring B₁] [Field A₂] [Ring B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂) (e₂ : B₁ ≃+* B₂) (he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)) {x : B₁} (h : IsSeparable A₁ x) : IsSeparable A₂ (e₂ x)

theorem Algebra.IsSeparable.of_equiv_equiv {A₁ : Type u_1} {B₁ : Type u_2} {A₂ : Type u_3} {B₂ : Type u_4} [Field A₁] [Ring B₁] [Field A₂] [Ring B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂) (e₂ : B₁ ≃+* B₂) (he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)) [Algebra.IsSeparable A₁ B₁] : Algebra.IsSeparable A₂ B₂

theorem AlgHom.natCard_of_powerBasis {S : Type u_2} [CommRing S] {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K S] [Algebra K L] (pb : PowerBasis K S) (h_sep : IsSeparable K pb.gen) (h_splits : Polynomial.Splits (algebraMap K L) (minpoly K pb.gen)) : Nat.card (S →ₐ[K] L) = pb.dim

theorem AlgHom.card_of_powerBasis {S : Type u_2} [CommRing S] {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K S] [Algebra K L] (pb : PowerBasis K S) (h_sep : IsSeparable K pb.gen) (h_splits : Polynomial.Splits (algebraMap K L) (minpoly K pb.gen)) : Fintype.card (S →ₐ[K] L) = pb.dim