Primitive Element Theorem #

In this file we prove the primitive element theorem.

Main results #

exists_primitive_element: a finite separable extension E / F has a primitive element, i.e. there is an α : E such that F⟮α⟯ = (⊤ : Subalgebra F E).

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 #

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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, F⟮α⟯ = ⊤

Primitive element theorem assuming E is finite.

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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, F⟮α⟯ = ⊤

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

Primitive element theorem for infinite fields #

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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

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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, F⟮α, β⟯ = F⟮γ⟯

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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, F⟮α⟯ = ⊤

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).

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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] :

PowerBasis F E

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