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/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import linear_algebra.matrix.charpoly.coeff
import linear_algebra.matrix.to_lin
import ring_theory.power_basis
/-!
# The minimal polynomial divides the characteristic polynomial of a matrix.
-/
noncomputable theory
universes u v
open polynomial matrix
variables {R : Type u} [comm_ring R]
variables {n : Type v} [decidable_eq n] [fintype n]
open finset
variable {M : matrix n n R}
namespace matrix
theorem is_integral : is_integral R M := ⟨M.charpoly, ⟨charpoly_monic M, aeval_self_charpoly M⟩⟩
theorem minpoly_dvd_charpoly {K : Type*} [field K] (M : matrix n n K) :
(minpoly K M) ∣ M.charpoly :=
minpoly.dvd _ _ (aeval_self_charpoly M)
end matrix
section power_basis
open algebra
/-- The characteristic polynomial of the map `λ x, a * x` is the minimal polynomial of `a`.
In combination with `det_eq_sign_charpoly_coeff` or `trace_eq_neg_charpoly_coeff`
and a bit of rewriting, this will allow us to conclude the
field norm resp. trace of `x` is the product resp. sum of `x`'s conjugates.
-/
lemma charpoly_left_mul_matrix {K S : Type*} [field K] [comm_ring S] [algebra K S]
(h : power_basis K S) :
(left_mul_matrix h.basis h.gen).charpoly = minpoly K h.gen :=
begin
apply minpoly.unique,
{ apply matrix.charpoly_monic },
{ apply (injective_iff_map_eq_zero (left_mul_matrix _)).mp (left_mul_matrix_injective h.basis),
rw [← polynomial.aeval_alg_hom_apply, aeval_self_charpoly] },
{ intros q q_monic root_q,
rw [matrix.charpoly_degree_eq_dim, fintype.card_fin, degree_eq_nat_degree q_monic.ne_zero],
apply with_bot.some_le_some.mpr,
exact h.dim_le_nat_degree_of_root q_monic.ne_zero root_q }
end
end power_basis
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