Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

[Merged by Bors] - feat(field_theory): state primitive element theorem using power_basis #8073

Closed
wants to merge 2 commits into from
Closed

[Merged by Bors] - feat(field_theory): state primitive element theorem using power_basis #8073

Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
76ef1dd
feat(field_theory): state primitive element theorem using `power_basis`
Vierkantor Jun 23, 2021
4561dfc
Does this work?
Vierkantor Jun 24, 2021
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
12 changes: 12 additions & 0 deletions src/field_theory/primitive_element.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -184,4 +184,16 @@ begin
exact induction_on_adjoin P base ih ⊤ },
end

/-- 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`. -/
noncomputable def power_basis_of_finite_of_separable
[finite_dimensional F E] (F_sep : is_separable F E) :
power_basis F E :=
let α := (exists_primitive_element F_sep).some,
pb := (adjoin.power_basis (F_sep.is_integral α)) in
have e : F⟮α⟯ = ⊤ := (exists_primitive_element F_sep).some_spec,
pb.map ((intermediate_field.equiv_of_eq e).trans intermediate_field.top_equiv)

end field