mathlib documentation

field_theory.primitive_element

Primitive Element Theorem #

In this file we prove the primitive element theorem.

Main results #

Implementation notes #

In declaration names, primitive_element abbreviates adjoin_simple_eq_top: it stands for the statement F⟮α⟯ = (⊤ : subalgebra F E). We did not add an extra declaration is_primitive_element F α := F⟮α⟯ = (⊤ : subalgebra F E) because this requires more unfolding without much obvious benefit.

Tags #

primitive element, separable field extension, separable extension, intermediate field, adjoin, exists_adjoin_simple_eq_top

Primitive element theorem for finite fields #

theorem field.exists_primitive_element_of_fintype_top (F : Type u_1) [field F] (E : Type u_2) [field E] [algebra F E] [fintype E] :
∃ (α : E), Fα =

Primitive element theorem assuming E is finite.

theorem field.exists_primitive_element_of_fintype_bot (F : Type u_1) [field F] (E : Type u_2) [field E] [algebra F E] [fintype F] [finite_dimensional F E] :
∃ (α : E), Fα =

Primitive element theorem for finite dimensional extension of a finite field.

Primitive element theorem for infinite fields #

theorem field.primitive_element_inf_aux_exists_c {F : Type u_1} [field F] [infinite F] {E : Type u_2} [field E] (ϕ : F →+* E) (α β : E) (f g : polynomial F) :
∃ (c : F), ∀ (α' : E), α' (polynomial.map ϕ f).roots∀ (β' : E), β' (polynomial.map ϕ g).roots-(α' - α) / (β' - β) ϕ c
theorem field.primitive_element_inf_aux (F : Type u_1) [field F] [infinite F] {E : Type u_2} [field E] (α β : E) [algebra F E] [is_separable F E] :
∃ (γ : E), Fα, β = (Fγ)
theorem field.exists_primitive_element (F : Type u_1) (E : Type u_2) [field F] [field E] [algebra F E] [finite_dimensional F E] [is_separable F E] :
∃ (α : E), Fα =

Primitive element theorem: a finite separable field extension E of F has a primitive element, i.e. there is an α ∈ E such that F⟮α⟯ = (⊤ : subalgebra F E).

def field.power_basis_of_finite_of_separable (F : Type u_1) (E : Type u_2) [field F] [field E] [algebra F E] [finite_dimensional F E] [is_separable F E] :

Alternative phrasing of primitive element theorem: a finite separable field extension has a basis 1, α, α^2, ..., α^n.

See also exists_primitive_element.

Equations
@[instance]
def field.alg_hom.fintype (F : Type u_1) (E : Type u_2) [field F] [field E] [algebra F E] [finite_dimensional F E] [is_separable F E] {K : Type u_3} [field K] [algebra F K] :

If E / F is a finite separable extension, then there are finitely many embeddings from E into K that fix F, corresponding to the number of conjugate roots of the primitive element generating F.

Equations
@[simp]
theorem alg_hom.card (F : Type u_1) (E : Type u_2) (K : Type u_3) [field F] [field E] [field K] [is_alg_closed K] [algebra F E] [finite_dimensional F E] [is_separable F E] [algebra F K] :