mathlib documentation

field_theory.splitting_field

Splitting fields #

This file introduces the notion of a splitting field of a polynomial and provides an embedding from a splitting field to any field that splits the polynomial. A polynomial f : polynomial K splits over a field extension L of K if it is zero or all of its irreducible factors over L have degree 1. A field extension of K of a polynomial f : polynomial K is called a splitting field if it is the smallest field extension of K such that f splits.

Main definitions #

Main statements #

def polynomial.splits {K : Type v} {L : Type w} [comm_ring 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.

Equations
@[simp]
theorem polynomial.splits_zero {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) :
theorem polynomial.splits_of_map_eq_C {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} {a : L} (h : polynomial.map i f = polynomial.C a) :
@[simp]
theorem polynomial.splits_C {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) (a : K) :
theorem polynomial.splits_of_map_degree_eq_one {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : (polynomial.map i f).degree = 1) :
theorem polynomial.splits_of_degree_le_one {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : f.degree 1) :
theorem polynomial.splits_of_degree_eq_one {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : f.degree = 1) :
theorem polynomial.splits_of_nat_degree_le_one {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : f.nat_degree 1) :
theorem polynomial.splits_of_nat_degree_eq_one {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : f.nat_degree = 1) :
theorem polynomial.splits_mul {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f g : polynomial K} (hf : polynomial.splits i f) (hg : polynomial.splits i g) :
theorem polynomial.splits_of_splits_mul' {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f g : polynomial K} (hfg : polynomial.map i (f * g) 0) (h : polynomial.splits i (f * g)) :
theorem polynomial.splits_map_iff {F : Type u} {K : Type v} {L : Type w} [comm_ring K] [field L] [field F] (i : K →+* L) (j : L →+* F) {f : polynomial K} :
theorem polynomial.splits_one {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) :
theorem polynomial.splits_of_is_unit {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) [is_domain K] {u : polynomial K} (hu : is_unit u) :
theorem polynomial.splits_X_sub_C {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {x : K} :
theorem polynomial.splits_X {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) :
theorem polynomial.splits_prod {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {ι : Type u} {s : ι → polynomial K} {t : finset ι} :
(∀ (j : ι), j tpolynomial.splits i (s j))polynomial.splits i (t.prod (λ (x : ι), s x))
theorem polynomial.splits_pow {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : polynomial.splits i f) (n : ) :
theorem polynomial.splits_X_pow {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) (n : ) :
theorem polynomial.splits_id_iff_splits {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} :
theorem polynomial.exists_root_of_splits' {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hs : polynomial.splits i f) (hf0 : (polynomial.map i f).degree 0) :
∃ (x : L), polynomial.eval₂ i x f = 0
theorem polynomial.roots_ne_zero_of_splits' {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hs : polynomial.splits i f) (hf0 : (polynomial.map i f).nat_degree 0) :
noncomputable def polynomial.root_of_splits' {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : polynomial.splits i f) (hfd : (polynomial.map i f).degree 0) :
L

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

Equations
theorem polynomial.map_root_of_splits' {K : Type v} {L : Type w} [comm_ring K] [field L] (i : K →+* L) {f : polynomial K} (hf : polynomial.splits i f) (hfd : (polynomial.map i f).degree 0) :
theorem polynomial.nat_degree_eq_card_roots' {K : Type v} {L : Type w} [comm_ring K] [field L] {p : polynomial K} {i : K →+* L} (hsplit : polynomial.splits i p) :
theorem polynomial.degree_eq_card_roots' {K : Type v} {L : Type w} [comm_ring K] [field L] {p : polynomial K} {i : K →+* L} (p_ne_zero : polynomial.map i p 0) (hsplit : polynomial.splits i p) :
theorem polynomial.splits_iff {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) (f : polynomial K) :
polynomial.splits i f f = 0 ∀ {g : polynomial L}, irreducible gg polynomial.map i fg.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 : polynomial.splits i f) :
f = 0 ∀ {g : polynomial L}, irreducible gg polynomial.map i fg.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 : polynomial.splits i (f * 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 : polynomial.splits i f) (hgf : g f) :
theorem polynomial.splits_of_splits_gcd_left {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f g : polynomial K} (hf0 : f 0) (hf : polynomial.splits i f) :
theorem polynomial.splits_of_splits_gcd_right {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f g : polynomial K} (hg0 : g 0) (hg : polynomial.splits i 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) :
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 : ι), j ts j 0)(polynomial.splits i (t.prod (λ (x : ι), s x)) ∀ (j : ι), j tpolynomial.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 : polynomial.splits (ring_hom.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 : polynomial.splits i f) (hf0 : f.degree 0) :
∃ (x : L), polynomial.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 : polynomial.splits i f) (hf0 : f.nat_degree 0) :
noncomputable def polynomial.root_of_splits {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f : polynomial K} (hf : polynomial.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.

Equations
theorem polynomial.root_of_splits'_eq_root_of_splits {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f : polynomial K} (hf : polynomial.splits i f) (hfd : (polynomial.map i f).degree 0) :

root_of_splits' is definitionally equal to root_of_splits.

theorem polynomial.map_root_of_splits {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f : polynomial K} (hf : polynomial.splits i f) (hfd : f.degree 0) :
theorem polynomial.nat_degree_eq_card_roots {K : Type v} {L : Type w} [field K] [field L] {p : polynomial K} {i : K →+* L} (hsplit : polynomial.splits i p) :
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 : polynomial.splits i p) :
theorem polynomial.roots_map {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f : polynomial K} (hf : polynomial.splits (ring_hom.id K) f) :
theorem polynomial.image_root_set {F : Type u} {K : Type v} {L : Type w} [field K] [field L] [field F] [algebra F K] [algebra F L] {p : polynomial F} (h : polynomial.splits (algebra_map F K) p) (f : K →ₐ[F] L) :
theorem polynomial.adjoin_root_set_eq_range {F : Type u} {K : Type v} {L : Type w} [field K] [field L] [field F] [algebra F K] [algebra F L] {p : polynomial F} (h : polynomial.splits (algebra_map F K) p) (f : K →ₐ[F] L) :
theorem polynomial.eq_prod_roots_of_splits {K : Type v} {L : Type w} [field K] [field L] {p : polynomial K} {i : K →+* L} (hsplit : polynomial.splits i p) :
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 : polynomial.splits i h) (h_roots : (polynomial.map i h).roots = {i x}) :
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 : polynomial.map i f = polynomial.C (i f.leading_coeff) * (multiset.map (λ (a : L), polynomial.X - polynomial.C a) s).prod) :
theorem polynomial.splits_of_splits_id {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f : polynomial K} :
theorem polynomial.splits_iff_exists_multiset {K : Type v} {L : Type w} [field K] [field L] (i : K →+* L) {f : polynomial K} :
theorem polynomial.splits_comp_of_splits {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 : polynomial.splits i f) :

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] {P : polynomial K} (hmo : P.monic) (hP : polynomial.splits (algebra_map K L) P) {r : L} (hr : r (polynomial.map (algebra_map K L) P).roots) :
theorem polynomial.prod_roots_eq_coeff_zero_of_monic_of_split {K : Type v} [field K] {P : polynomial K} (hmo : P.monic) (hP : polynomial.splits (ring_hom.id K) P) :
P.coeff 0 = (-1) ^ P.nat_degree * P.roots.prod

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

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

noncomputable def alg_equiv.adjoin_singleton_equiv_adjoin_root_minpoly (F : Type u) [field F] {R : Type u_1} [comm_ring R] [algebra F R] (x : R) :

If p is the minimal polynomial of a over F then F[a] ≃ₐ[F] F[x]/(p)

Equations
theorem finite_dimensional.of_subalgebra_to_submodule {K : Type u_1} {V : Type u_2} [field K] [ring V] [algebra K V] {s : subalgebra K V} (h : finite_dimensional K (s.to_submodule)) :

If a subalgebra is finite_dimensional as a submodule then it is finite_dimensional.

theorem lift_of_splits {F : Type u_1} {K : Type u_2} {L : Type u_3} [field F] [field K] [field L] [algebra F K] [algebra F L] (s : finset K) :
(∀ (x : K), x sis_integral F x polynomial.splits (algebra_map F L) (minpoly F x))nonempty ((algebra.adjoin F s) →ₐ[F] L)

If K and L are field extensions of F and we have s : finset K such that the minimal polynomial of each x ∈ s splits in L then algebra.adjoin F s embeds in L.

noncomputable def polynomial.factor {K : Type v} [field K] (f : polynomial K) :

Non-computably choose an irreducible factor from a polynomial.

Equations
theorem polynomial.irreducible_factor {K : Type v} [field K] (f : polynomial K) :
theorem polynomial.factor_dvd_of_not_is_unit {K : Type v} [field K] {f : polynomial K} (hf1 : ¬is_unit f) :
theorem polynomial.factor_dvd_of_degree_ne_zero {K : Type v} [field K] {f : polynomial K} (hf : f.degree 0) :
theorem polynomial.factor_dvd_of_nat_degree_ne_zero {K : Type v} [field K] {f : polynomial K} (hf : f.nat_degree 0) :
noncomputable def polynomial.remove_factor {K : Type v} [field K] (f : polynomial K) :

Divide a polynomial f by X - C r where r is a root of f in a bigger field extension.

Equations
theorem polynomial.nat_degree_remove_factor' {K : Type v} [field K] {f : polynomial K} {n : } (hfn : f.nat_degree = n + 1) :
def polynomial.splitting_field_aux (n : ) {K : Type u} [field K] (f : polynomial K) :
Type u

Auxiliary construction to a splitting field of a polynomial, which removes n (arbitrarily-chosen) factors.

Uses recursion on the degree. For better definitional behaviour, structures including splitting_field_aux (such as instances) should be defined using this recursion in each field, rather than defining the whole tuple through recursion.

Equations
Instances for polynomial.splitting_field_aux
@[protected, instance]
noncomputable def polynomial.splitting_field_aux.field (n : ) {K : Type u} [field K] {f : polynomial K} :
Equations
@[protected, instance]
noncomputable def polynomial.splitting_field_aux.inhabited {K : Type v} [field K] {n : } {f : polynomial K} :
Equations
@[protected, instance]
noncomputable def polynomial.splitting_field_aux.algebra (n : ) (R : Type u_1) {K : Type u} [comm_semiring R] [field K] [algebra R K] {f : polynomial K} :
Equations
@[protected, instance]
def polynomial.splitting_field_aux.is_scalar_tower (n : ) (R₁ : Type u_1) (R₂ : Type u_2) {K : Type u} [comm_semiring R₁] [comm_semiring R₂] [has_smul R₁ R₂] [field K] [algebra R₁ K] [algebra R₂ K] [is_scalar_tower R₁ R₂ K] {f : polynomial K} :
@[protected, instance]
@[protected]
theorem polynomial.splitting_field_aux.exists_lift (n : ) {K : Type u} [field K] (f : polynomial K) (hfn : f.nat_degree = n) {L : Type u_1} [field L] (j : K →+* L) (hf : polynomial.splits j f) :
@[protected, instance]
noncomputable def polynomial.splitting_field.inhabited {K : Type v} [field K] (f : polynomial K) :
Equations
@[protected, instance]
noncomputable def polynomial.splitting_field.algebra' {K : Type v} [field K] (f : polynomial K) {R : Type u_1} [comm_semiring R] [algebra R K] :

This should be an instance globally, but it creates diamonds with the , , and algebras (via their smul and to_fun fields):

example :
  (algebra_nat : algebra  (splitting_field f)) = splitting_field.algebra' f :=
rfl  -- fails

example :
  (algebra_int _ : algebra  (splitting_field f)) = splitting_field.algebra' f :=
rfl  -- fails

example [char_zero K] [char_zero (splitting_field f)] :
  (algebra_rat : algebra  (splitting_field f)) = splitting_field.algebra' f :=
rfl  -- fails

Until we resolve these diamonds, it's more convenient to only turn this instance on with local attribute [instance] in places where the benefit of having the instance outweighs the cost.

In the meantime, the splitting_field.algebra instance below is immune to these particular diamonds since K = ℕ and K = ℤ are not possible due to the field K assumption. Diamonds in algebra ℚ (splitting_field f) instances are still possible via this instance unfortunately, but these are less common as they require suitable char_zero instances to be present.

Equations
@[protected]
noncomputable def polynomial.splitting_field.lift {K : Type v} {L : Type w} [field K] [field L] (f : polynomial K) [algebra K L] (hb : polynomial.splits (algebra_map K L) f) :

Embeds the splitting field into any other field that splits the polynomial.

Equations
@[protected, instance]
def polynomial.is_splitting_field.map {F : Type u} {K : Type v} {L : Type w} [field K] [field L] [field F] [algebra K L] [algebra F K] [algebra F L] [is_scalar_tower F K L] (f : polynomial F) [polynomial.is_splitting_field F L f] :
theorem polynomial.is_splitting_field.mul {F : Type u} {K : Type v} (L : Type w) [field K] [field L] [field F] [algebra K L] [algebra F K] [algebra F L] [is_scalar_tower F K L] (f g : polynomial F) (hf : f 0) (hg : g 0) [polynomial.is_splitting_field F K f] [polynomial.is_splitting_field K L (polynomial.map (algebra_map F K) g)] :
noncomputable def polynomial.is_splitting_field.lift {F : Type u} {K : Type v} (L : Type w) [field K] [field L] [field F] [algebra K L] [algebra K F] (f : polynomial K) [polynomial.is_splitting_field K L f] (hf : polynomial.splits (algebra_map K F) f) :

Splitting field of f embeds into any field that splits f.

Equations
noncomputable def polynomial.is_splitting_field.alg_equiv {K : Type v} (L : Type w) [field K] [field L] [algebra K L] (f : polynomial K) [polynomial.is_splitting_field K L f] :

Any splitting field is isomorphic to splitting_field f.

Equations
theorem polynomial.is_splitting_field.of_alg_equiv {F : Type u} {K : Type v} (L : Type w) [field K] [field L] [field F] [algebra K L] [algebra K F] (p : polynomial K) (f : F ≃ₐ[K] L) [polynomial.is_splitting_field K F p] :
theorem intermediate_field.splits_of_splits {K : Type v} {L : Type w} [field K] [field L] [algebra K L] {p : polynomial K} {F : intermediate_field K L} (h : polynomial.splits (algebra_map K L) p) (hF : ∀ (x : L), x p.root_set Lx F) :