field_theory.minpoly.field

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theorem minpoly.degree_le_of_ne_zero (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B) {p : polynomial A} (pnz : p ≠ 0) (hp : ⇑(polynomial.aeval x) p = 0) :

(minpoly A x).degree ≤ p.degree

If an element x is a root of a nonzero polynomial p, then the degree of p is at least the degree of the minimal polynomial of x. See also gcd_domain_degree_le_of_ne_zero which relaxes the assumptions on A in exchange for stronger assumptions on B.

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theorem minpoly.ne_zero_of_finite_field_extension (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (e : B) [finite_dimensional A B] :

minpoly A e ≠ 0

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theorem minpoly.unique (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B) {p : polynomial A} (pmonic : p.monic) (hp : ⇑(polynomial.aeval x) p = 0) (pmin : ∀ (q : polynomial A), q.monic → ⇑(polynomial.aeval x) q = 0 → p.degree ≤ q.degree) :

p = minpoly A x

The minimal polynomial of an element x is uniquely characterized by its defining property: if there is another monic polynomial of minimal degree that has x as a root, then this polynomial is equal to the minimal polynomial of x. See also minpoly.gcd_unique which relaxes the assumptions on A in exchange for stronger assumptions on B.

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theorem minpoly.dvd (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B) {p : polynomial A} (hp : ⇑(polynomial.aeval x) p = 0) :

minpoly A x ∣ p

If an element x is a root of a polynomial p, then the minimal polynomial of x divides p. See also minpoly.gcd_domain_dvd which relaxes the assumptions on A in exchange for stronger assumptions on B.

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theorem minpoly.dvd_map_of_is_scalar_tower (A : Type u_1) (K : Type u_2) {R : Type u_3} [comm_ring A] [field K] [comm_ring R] [algebra A K] [algebra A R] [algebra K R] [is_scalar_tower A K R] (x : R) :

minpoly K x ∣ polynomial.map (algebra_map A K) (minpoly A x)

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theorem minpoly.dvd_map_of_is_scalar_tower' (R : Type u_1) {S : Type u_2} (K : Type u_3) (L : Type u_4) [comm_ring R] [comm_ring S] [field K] [comm_ring L] [algebra R S] [algebra R K] [algebra S L] [algebra K L] [algebra R L] [is_scalar_tower R K L] [is_scalar_tower R S L] (s : S) :

minpoly K (⇑(algebra_map S L) s) ∣ polynomial.map (algebra_map R K) (minpoly R s)

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theorem minpoly.aeval_of_is_scalar_tower (R : Type u_1) {K : Type u_2} {T : Type u_3} {U : Type u_4} [comm_ring R] [field K] [comm_ring T] [algebra R K] [algebra K T] [algebra R T] [is_scalar_tower R K T] [comm_semiring U] [algebra K U] [algebra R U] [is_scalar_tower R K U] (x : T) (y : U) (hy : ⇑(polynomial.aeval y) (minpoly K x) = 0) :

⇑(polynomial.aeval y) (minpoly R x) = 0

If y is a conjugate of x over a field K, then it is a conjugate over a subring R.

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theorem minpoly.eq_of_irreducible_of_monic {A : Type u_1} {B : Type u_2} [field A] [ring B] [algebra A B] {x : B} [nontrivial B] {p : polynomial A} (hp1 : irreducible p) (hp2 : ⇑(polynomial.aeval x) p = 0) (hp3 : p.monic) :

p = minpoly A x

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theorem minpoly.eq_of_irreducible {A : Type u_1} {B : Type u_2} [field A] [ring B] [algebra A B] {x : B} [nontrivial B] {p : polynomial A} (hp1 : irreducible p) (hp2 : ⇑(polynomial.aeval x) p = 0) :

p * ⇑polynomial.C (p.leading_coeff)⁻¹ = minpoly A x

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theorem minpoly.eq_of_algebra_map_eq {K : Type u_1} {S : Type u_2} {T : Type u_3} [field K] [comm_ring S] [comm_ring T] [algebra K S] [algebra K T] [algebra S T] [is_scalar_tower K S T] (hST : function.injective ⇑(algebra_map S T)) {x : S} {y : T} (hx : is_integral K x) (h : y = ⇑(algebra_map S T) x) :

minpoly K x = minpoly K y

If y is the image of x in an extension, their minimal polynomials coincide.

We take h : y = algebra_map L T x as an argument because rw h typically fails since is_integral R y depends on y.

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theorem minpoly.add_algebra_map {A : Type u_1} [field A] {B : Type u_2} [comm_ring B] [algebra A B] {x : B} (hx : is_integral A x) (a : A) :

minpoly A (x + ⇑(algebra_map A B) a) = (minpoly A x).comp (polynomial.X - ⇑polynomial.C a)

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theorem minpoly.sub_algebra_map {A : Type u_1} [field A] {B : Type u_2} [comm_ring B] [algebra A B] {x : B} (hx : is_integral A x) (a : A) :

minpoly A (x - ⇑(algebra_map A B) a) = (minpoly A x).comp (polynomial.X + ⇑polynomial.C a)

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noncomputable def minpoly.fintype.subtype_prod {E : Type u_1} {X : set E} (hX : X.finite) {L : Type u_2} (F : E → multiset L) :

fintype (Π (x : ↥X), {l // l ∈ F ↑x})

A technical finiteness result.

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minpoly.fintype.subtype_prod hX F = let hX : fintype ↥X := hX.fintype in pi.fintype

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def minpoly.roots_of_min_poly_pi_type (F : Type u_3) (E : Type u_4) (K : Type u_5) [field F] [ring E] [comm_ring K] [is_domain K] [algebra F E] [algebra F K] [finite_dimensional F E] (φ : E →ₐ[F] K) (x : ↥(set.range ⇑(finite_dimensional.fin_basis F E))) :

{l // l ∈ (polynomial.map (algebra_map F K) (minpoly F x.val)).roots}

Function from Hom_K(E,L) to pi type Π (x : basis), roots of min poly of x

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minpoly.roots_of_min_poly_pi_type F E K φ x = ⟨⇑φ ↑x, _⟩

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theorem minpoly.aux_inj_roots_of_min_poly (F : Type u_3) (E : Type u_4) (K : Type u_5) [field F] [ring E] [comm_ring K] [is_domain K] [algebra F E] [algebra F K] [finite_dimensional F E] :

function.injective (minpoly.roots_of_min_poly_pi_type F E K)

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@[protected, instance]

noncomputable def minpoly.alg_hom.fintype (F : Type u_3) (E : Type u_4) (K : Type u_5) [field F] [ring E] [comm_ring K] [is_domain K] [algebra F E] [algebra F K] [finite_dimensional F E] :

fintype (E →ₐ[F] K)

Given field extensions E/F and K/F, with E/F finite, there are finitely many F-algebra homomorphisms E →ₐ[K] K.

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minpoly.alg_hom.fintype F E K = fintype.of_injective (minpoly.roots_of_min_poly_pi_type F E K) _

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theorem minpoly.eq_X_sub_C {A : Type u_1} (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] (a : A) :

minpoly A (⇑(algebra_map A B) a) = polynomial.X - ⇑polynomial.C a

If B/K is a nontrivial algebra over a field, and x is an element of K, then the minimal polynomial of algebra_map K B x is X - C x.

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theorem minpoly.eq_X_sub_C' {A : Type u_1} [field A] (a : A) :

minpoly A a = polynomial.X - ⇑polynomial.C a

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@[simp]

theorem minpoly.zero (A : Type u_1) (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] :

minpoly A 0 = polynomial.X

The minimal polynomial of 0 is X.

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@[simp]

theorem minpoly.one (A : Type u_1) (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] :

minpoly A 1 = polynomial.X - 1

The minimal polynomial of 1 is X - 1.

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theorem minpoly.prime {A : Type u_1} {B : Type u_2} [field A] [ring B] [is_domain B] [algebra A B] {x : B} (hx : is_integral A x) :

prime (minpoly A x)

A minimal polynomial is prime.

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theorem minpoly.root {A : Type u_1} {B : Type u_2} [field A] [ring B] [is_domain B] [algebra A B] {x : B} (hx : is_integral A x) {y : A} (h : (minpoly A x).is_root y) :

⇑(algebra_map A B) y = x

If L/K is a field extension and an element y of K is a root of the minimal polynomial of an element x ∈ L, then y maps to x under the field embedding.

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@[simp]

theorem minpoly.coeff_zero_eq_zero {A : Type u_1} {B : Type u_2} [field A] [ring B] [is_domain B] [algebra A B] {x : B} (hx : is_integral A x) :

(minpoly A x).coeff 0 = 0 ↔ x = 0

The constant coefficient of the minimal polynomial of x is 0 if and only if x = 0.

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theorem minpoly.coeff_zero_ne_zero {A : Type u_1} {B : Type u_2} [field A] [ring B] [is_domain B] [algebra A B] {x : B} (hx : is_integral A x) (h : x ≠ 0) :

(minpoly A x).coeff 0 ≠ 0

The minimal polynomial of a nonzero element has nonzero constant coefficient.

mathlib3 documentation

Minimal polynomials on an algebra over a field #