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] - refactor(field_theory/algebraic_closure): move complex.is_alg_closed #7344

Closed
wants to merge 2 commits into from
Closed

[Merged by Bors] - refactor(field_theory/algebraic_closure): move complex.is_alg_closed #7344

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
8785b3d
refactor(field_theory/algebraic_closure): move complex.is_alg_closed
semorrison Apr 23, 2021
3ba2fa2
Update src/analysis/complex/polynomial.lean
semorrison Apr 23, 2021
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Jump to
Jump to file
Failed to load files.
Diff view
Diff view
10 changes: 8 additions & 2 deletions src/analysis/complex/polynomial.lean
View file
Open in desktop
Expand Up @@ -3,13 +3,16 @@ Copyright (c) 2019 Chris Hughes All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import topology.algebra.polynomial
import analysis.special_functions.pow
import field_theory.algebraic_closure
import topology.algebra.polynomial

/-!
# The fundamental theorem of algebra

This file proves that every nonconstant complex polynomial has a root.

As a consequence, the complex numbers are algebraically closed.
-/

open complex polynomial metric filter is_absolute_value set
Expand All @@ -21,7 +24,7 @@ namespace complex
<https://ncatlab.org/nlab/show/fundamental+theorem+of+algebra#classical_fta_via_advanced_calculus>
-/
/-- The fundamental theorem of algebra. Every non constant complex polynomial
has a root -/
has a root. -/
lemma exists_root {f : polynomial ℂ} (hf : 0 < degree f) : ∃ z : ℂ, is_root f z :=
let ⟨z₀, hz₀⟩ := f.exists_forall_norm_le in
exists.intro z₀ $ classical.by_contradiction $ λ hf0,
Expand Down Expand Up @@ -92,4 +95,7 @@ lt_irrefl (f.eval z₀).abs $
add_lt_add_of_le_of_lt (by rw hF₂) hF₃
... = (f.eval z₀).abs : sub_add_cancel _ _

instance is_alg_closed : is_alg_closed ℂ :=
is_alg_closed.of_exists_root _ $ λ p _ hp, complex.exists_root $ degree_pos_of_irreducible hp

end complex
4 changes: 0 additions & 4 deletions src/field_theory/algebraic_closure.lean
View file
Open in desktop
Expand Up @@ -6,7 +6,6 @@ Authors: Kenny Lau

import algebra.direct_limit
import field_theory.splitting_field
import analysis.complex.polynomial

/-!
# Algebraic Closure
Expand Down Expand Up @@ -109,9 +108,6 @@ algebra_map_surjective_of_is_integral ((is_algebraic_iff_is_integral' k).mp hf)

end is_alg_closed

instance complex.is_alg_closed : is_alg_closed ℂ :=
is_alg_closed.of_exists_root _ $ λ p _ hp, complex.exists_root $ degree_pos_of_irreducible hp

/-- Typeclass for an extension being an algebraic closure. -/
class is_alg_closure (K : Type v) [field K] [algebra k K] : Prop :=
(alg_closed : is_alg_closed K)
Expand Down