Primitive Element Theorem #

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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 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 #

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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 : E), F⟮α⟯ = ⊤

Primitive element theorem assuming E is finite.

source

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] [finite_dimensional F E] :

∃ (α : E), 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 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

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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 β : E) [algebra F E] [is_separable F E] :

∃ (γ : E), 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] [finite_dimensional F E] [is_separable F E] :

∃ (α : E), 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.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] :

power_basis F E

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