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Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Minimal polynomials over a GCD monoid #

This file specializes the theory of minpoly to the case of an algebra over a GCD monoid.

Main results #

theorem minpoly.isIntegrallyClosed_eq_field_fractions {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] (K : Type u_3) (L : Type u_4) [Field K] [Algebra R K] [IsFractionRing R K] [CommRing L] [Nontrivial L] [Algebra R L] [Algebra S L] [Algebra K L] [IsScalarTower R K L] [IsScalarTower R S L] [IsIntegrallyClosed R] [IsDomain S] {s : S} (hs : IsIntegral R s) :

For integrally closed domains, the minimal polynomial over the ring is the same as the minimal polynomial over the fraction field. See minpoly.isIntegrallyClosed_eq_field_fractions' if S is already a K-algebra.

theorem minpoly.isIntegrallyClosed_eq_field_fractions' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] (K : Type u_3) [Field K] [Algebra R K] [IsFractionRing R K] [IsIntegrallyClosed R] [IsDomain S] [Algebra K S] [IsScalarTower R K S] {s : S} (hs : IsIntegral R s) :

For integrally closed domains, the minimal polynomial over the ring is the same as the minimal polynomial over the fraction field. Compared to minpoly.isIntegrallyClosed_eq_field_fractions, this version is useful if the element is in a ring that is already a K-algebra.

theorem minpoly.isIntegrallyClosed_dvd {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {s : S} (hs : IsIntegral R s) {p : Polynomial R} (hp : (Polynomial.aeval s) p = 0) :
minpoly R s p

For integrally closed rings, the minimal polynomial divides any polynomial that has the integral element as root. See also minpoly.dvd which relaxes the assumptions on S in exchange for stronger assumptions on R.

theorem minpoly.isIntegrallyClosed_dvd_iff {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {s : S} (hs : IsIntegral R s) (p : Polynomial R) :
theorem minpoly.ker_eval {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {s : S} (hs : IsIntegral R s) :
theorem minpoly.IsIntegrallyClosed.degree_le_of_ne_zero {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {s : S} (hs : IsIntegral R s) {p : Polynomial R} (hp0 : p 0) (hp : (Polynomial.aeval s) p = 0) :
(minpoly R s).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 minpoly.degree_le_of_ne_zero which relaxes the assumptions on S in exchange for stronger assumptions on R.

theorem IsIntegrallyClosed.minpoly.unique {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {s : S} {P : Polynomial R} (hmo : P.Monic) (hP : (Polynomial.aeval s) P = 0) (Pmin : ∀ (Q : Polynomial R), Q.Monic(Polynomial.aeval s) Q = 0P.degree Q.degree) :
P = minpoly R s

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.unique which relaxes the assumptions on S in exchange for stronger assumptions on R.

theorem minpoly.prime_of_isIntegrallyClosed {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) :
@[simp]
theorem minpoly.equivAdjoin_symm_apply {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) (b : (Algebra.adjoin R {x})) :
def minpoly.equivAdjoin {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) :

The algebra isomorphism AdjoinRoot (minpoly R x) ≃ₐ[R] adjoin R x

Equations
Instances For
    def Algebra.adjoin.powerBasis' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) :

    The PowerBasis of adjoin R {x} given by x. See Algebra.adjoin.powerBasis for a version over a field.

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    Instances For
      @[simp]
      theorem Algebra.adjoin.powerBasis'_dim {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) :
      (Algebra.adjoin.powerBasis' hx).dim = (minpoly R x).natDegree
      @[simp]
      theorem Algebra.adjoin.powerBasis'_gen {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) :
      (Algebra.adjoin.powerBasis' hx).gen = x,
      noncomputable def PowerBasis.ofGenMemAdjoin' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (B : PowerBasis R S) (hint : IsIntegral R x) (hx : B.gen Algebra.adjoin R {x}) :

      The power basis given by x if B.gen ∈ adjoin R {x}.

      Equations
      Instances For
        @[simp]
        theorem PowerBasis.ofGenMemAdjoin'_dim {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (B : PowerBasis R S) (hint : IsIntegral R x) (hx : B.gen Algebra.adjoin R {x}) :
        (B.ofGenMemAdjoin' hint hx).dim = (minpoly R x).natDegree
        @[simp]
        theorem PowerBasis.ofGenMemAdjoin'_gen {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsDomain S] [NoZeroSMulDivisors R S] [IsIntegrallyClosed R] {x : S} (B : PowerBasis R S) (hint : IsIntegral R x) (hx : B.gen Algebra.adjoin R {x}) :
        (B.ofGenMemAdjoin' hint hx).gen = x
        instance minpoly.instIsScalarTowerSubtypeMemSubringSubalgebraIntegralClosure {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (A : Subring K) :
        IsScalarTower (↥A) (↥(integralClosure (↥A) L)) L
        Equations
        • =
        theorem minpoly.ofSubring {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (A : Subring K) [IsIntegrallyClosed A] [IsFractionRing (↥A) K] (x : (integralClosure (↥A) L)) :
        Polynomial.map (algebraMap (↥A) K) (minpoly (↥A) x) = minpoly K x

        The minimal polynomial of x : L over K agrees with its minimal polynomial over the integrally closed subring A.