# Documentation

Mathlib.FieldTheory.Minpoly.Basic

# Minimal polynomials #

This file defines the minimal polynomial of an element x of an A-algebra B, under the assumption that x is integral over A, and derives some basic properties such as irreducibility under the assumption B is a domain.

noncomputable def minpoly (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) :

Suppose x : B, where B is an A-algebra.

The minimal polynomial minpoly A x of x is a monic polynomial with coefficients in A of smallest degree that has x as its root, if such exists (IsIntegral A x) or zero otherwise.

For example, if V is a 𝕜-vector space for some field 𝕜 and f : V →ₗ[𝕜] V then the minimal polynomial of f is minpoly 𝕜 f.

Instances For
theorem minpoly.monic {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] {x : B} (hx : ) :

A minimal polynomial is monic.

theorem minpoly.ne_zero {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] {x : B} [] (hx : ) :
minpoly A x 0

A minimal polynomial is nonzero.

theorem minpoly.eq_zero {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] {x : B} (hx : ¬) :
minpoly A x = 0
theorem minpoly.algHom_eq {A : Type u_1} {B : Type u_2} {B' : Type u_3} [] [Ring B] [Ring B'] [Algebra A B] [Algebra A B'] (f : B →ₐ[A] B') (hf : ) (x : B) :
minpoly A (f x) = minpoly A x
theorem minpoly.algebraMap_eq {A : Type u_1} {B' : Type u_3} [] [Ring B'] [Algebra A B'] {B : Type u_4} [] [Algebra A B] [Algebra B B'] [IsScalarTower A B B'] (h : Function.Injective ↑(algebraMap B B')) (x : B) :
minpoly A (↑(algebraMap B B') x) = minpoly A x
@[simp]
theorem minpoly.algEquiv_eq {A : Type u_1} {B : Type u_2} {B' : Type u_3} [] [Ring B] [Ring B'] [Algebra A B] [Algebra A B'] (f : B ≃ₐ[A] B') (x : B) :
minpoly A (f x) = minpoly A x
@[simp]
theorem minpoly.aeval (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) :
↑() (minpoly A x) = 0

An element is a root of its minimal polynomial.

theorem minpoly.ne_one (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) [] :
minpoly A x 1

A minimal polynomial is not 1.

theorem minpoly.map_ne_one (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) [] {R : Type u_4} [] [] (f : A →+* R) :
theorem minpoly.not_isUnit (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) [] :

A minimal polynomial is not a unit.

theorem minpoly.mem_range_of_degree_eq_one (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) (hx : Polynomial.degree (minpoly A x) = 1) :
theorem minpoly.min (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) {p : } (pmonic : ) (hp : ↑() p = 0) :

The defining property of the minimal polynomial of an element x: it is the monic polynomial with smallest degree that has x as its root.

theorem minpoly.unique' (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) {p : } (hm : ) (hp : ↑() p = 0) (hl : ∀ (q : ), q = 0 ↑() q 0) :
p = minpoly A x
theorem minpoly.subsingleton (A : Type u_1) {B : Type u_2} [] [Ring B] [Algebra A B] (x : B) [] :
minpoly A x = 1
theorem minpoly.natDegree_pos {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] {x : B} [] (hx : ) :
0 <

The degree of a minimal polynomial, as a natural number, is positive.

theorem minpoly.degree_pos {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] {x : B} [] (hx : ) :

The degree of a minimal polynomial is positive.

theorem minpoly.eq_X_sub_C_of_algebraMap_inj {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] (a : A) (hf : Function.Injective ↑()) :
minpoly A (↑() a) = Polynomial.X - Polynomial.C a

If B/A is an injective ring extension, and a is an element of A, then the minimal polynomial of algebraMap A B a is X - C a.

theorem minpoly.aeval_ne_zero_of_dvdNotUnit_minpoly {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] {x : B} {a : } (hx : ) (hamonic : ) (hdvd : DvdNotUnit a (minpoly A x)) :
↑() a 0

If a strictly divides the minimal polynomial of x, then x cannot be a root for a.

theorem minpoly.irreducible {A : Type u_1} {B : Type u_2} [] [Ring B] [Algebra A B] {x : B} [] [] (hx : ) :

A minimal polynomial is irreducible.