Documentation

Mathlib.FieldTheory.PrimitiveElement

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 IsPrimitiveElement 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_finite_top (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] [Finite E] :
α, Fα =

Primitive element theorem assuming E is finite.

theorem Field.exists_primitive_element_of_finite_bot (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] [Finite F] [FiniteDimensional F E] :
α, 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 : E) (β : E) (f : Polynomial F) (g : Polynomial F) :
c, ∀ (α' : E), α' Polynomial.roots (Polynomial.map ϕ f)∀ (β' : E), β' Polynomial.roots (Polynomial.map ϕ g)-(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 : E) (β : E) [Algebra F E] [IsSeparable F E] :
γ, Fα, β = Fγ
theorem Field.exists_primitive_element (F : Type u_1) (E : Type u_2) [Field F] [Field E] [Algebra F E] [FiniteDimensional F E] [IsSeparable F E] :
α, 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).

noncomputable def Field.powerBasisOfFiniteOfSeparable (F : Type u_1) (E : Type u_2) [Field F] [Field E] [Algebra F E] [FiniteDimensional F E] [IsSeparable 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.

Instances For
    @[simp]
    theorem AlgHom.card (F : Type u_1) (E : Type u_2) (K : Type u_3) [Field F] [Field E] [Field K] [IsAlgClosed K] [Algebra F E] [FiniteDimensional F E] [IsSeparable F E] [Algebra F K] :