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

• minpoly.isIntegrallyClosed_eq_field_fractions: For integrally closed domains, the minimal polynomial over the ring is the same as the minimal polynomial over the fraction field.

• minpoly.isIntegrallyClosed_dvd: For integrally closed domains, the minimal polynomial divides any primitive polynomial that has the integral element as root.

• IsIntegrallyClosed.Minpoly.unique: 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.

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) : minpoly K ((algebraMap S L) s) = Polynomial.map (algebraMap R K) (minpoly 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) : minpoly K s = Polynomial.map (algebraMap R K) (minpoly 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] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {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] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {s : S} (hs : IsIntegral R s) (p : Polynomial R) : (Polynomial.aeval s) p = 0 ↔ minpoly R s ∣ p

theorem minpoly.ker_eval {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {s : S} (hs : IsIntegral R s) : RingHom.ker (Polynomial.aeval s).toRingHom = Ideal.span {minpoly 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] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {s : 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 minpoly.IsIntegrallyClosed.isIntegral_iff_isUnit_leadingCoeff {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} {p : Polynomial R} (hirr : Irreducible p) (hp : (Polynomial.aeval x) p = 0) : IsIntegral R x ↔ IsUnit p.leadingCoeff

If x is a root of an irreducible polynomial p, then x is integral iff the leading coefficient of p is a unit.

theorem IsIntegrallyClosed.minpoly.unique {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {s : S} {P : Polynomial R} (hmo : P.Monic) (hP : (Polynomial.aeval s) P = 0) (Pmin : ∀ (Q : Polynomial R), Q.Monic → (Polynomial.aeval s) Q = 0 → P.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.IsIntegrallyClosed.unique_of_degree_le_degree_minpoly {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {s : S} {p : Polynomial R} (hmo : p.Monic) (hp : (Polynomial.aeval s) p = 0) (pmin : p.degree ≤ (minpoly R s).degree) : p = minpoly R s

theorem minpoly.IsIntegrallyClosed.isIntegral_iff_leadingCoeff_dvd {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {s : S} {p : Polynomial R} (hp : (Polynomial.aeval s) p = 0) (h₀ : p ≠ 0) (pmin : ∀ (q : Polynomial R), q.Monic → (Polynomial.aeval s) q = 0 → p.degree ≤ q.degree) : IsIntegral R s ↔ Polynomial.C p.leadingCoeff ∣ p

theorem minpoly.prime_of_isIntegrallyClosed {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : Prime (minpoly R x)

theorem IsIntegrallyClosed.minpoly_smul {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {r : R} (hr : r ≠ 0) {s : S} (hs : IsIntegral R s) : minpoly R (r • s) = (minpoly R s).scaleRoots r

theorem minpoly.ToAdjoin.injective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : Function.Injective ⇑(AdjoinRoot.Minpoly.toAdjoin R x)

def minpoly.equivAdjoin {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : AdjoinRoot (minpoly R x) ≃ₐ[R] ↥(Algebra.adjoin R {x})

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

@[simp]

theorem minpoly.equivAdjoin_toAlgHom {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : ↑(equivAdjoin hx) = AdjoinRoot.Minpoly.toAdjoin R x

@[simp]

theorem minpoly.coe_equivAdjoin {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : ⇑(equivAdjoin hx) = ⇑(AdjoinRoot.Minpoly.toAdjoin R x)

def Algebra.adjoin.powerBasis' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : PowerBasis R ↥(adjoin R {x})

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

@[simp]

theorem Algebra.adjoin.powerBasis'_dim {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : (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] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) : (powerBasis' hx).gen = ⟨x, ⋯⟩

noncomputable def PowerBasis.ofAdjoinEqTop' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) (hx' : Algebra.adjoin R {x} = ⊤) : PowerBasis R S

If x generates S over R and is integral over R, then it defines a power basis. See PowerBasis.ofAdjoinEqTop for a version over a field.

@[deprecated PowerBasis.ofAdjoinEqTop' "Use in combination with `PowerBasis.adjoin_eq_top_of_gen_mem_adjoin` to recover the deprecated definition" (since := "2025-09-28")]

def PowerBasis.ofGenMemAdjoin' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) (hx' : Algebra.adjoin R {x} = ⊤) : PowerBasis R S

Alias of PowerBasis.ofAdjoinEqTop'.

---

If x generates S over R and is integral over R, then it defines a power basis. See PowerBasis.ofAdjoinEqTop for a version over a field.

@[simp]

theorem PowerBasis.ofAdjoinEqTop'_dim {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) (hx' : Algebra.adjoin R {x} = ⊤) : (ofAdjoinEqTop' hx hx').dim = (minpoly R x).natDegree

@[simp]

theorem PowerBasis.ofAdjoinEqTop'_gen {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) (hx' : Algebra.adjoin R {x} = ⊤) : (ofAdjoinEqTop' hx hx').gen = x

@[deprecated PowerBasis.ofAdjoinEqTop'_dim "Use in combination with `PowerBasis.adjoin_eq_top_of_gen_mem_adjoin` to recover the deprecated definition" (since := "2025-09-28")]

theorem PowerBasis.ofGenMemAdjoin'_dim {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) (hx' : Algebra.adjoin R {x} = ⊤) : (ofAdjoinEqTop' hx hx').dim = (minpoly R x).natDegree

Alias of PowerBasis.ofAdjoinEqTop'_dim.

@[deprecated PowerBasis.ofAdjoinEqTop'_gen "Use in combination with `PowerBasis.adjoin_eq_top_of_gen_mem_adjoin` to recover the deprecated definition" (since := "2025-09-28")]

theorem PowerBasis.ofGenMemAdjoin'_gen {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain R] [Algebra R S] [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {x : S} (hx : IsIntegral R x) (hx' : Algebra.adjoin R {x} = ⊤) : (ofAdjoinEqTop' hx hx').gen = x

Alias of PowerBasis.ofAdjoinEqTop'_gen.

instance minpoly.instAlgebraSubtypeMemSubringSubalgebraIntegralClosure {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (A : Subring K) : Algebra ↥A ↥(integralClosure (↥A) L)

instance minpoly.instSMulSubtypeMemSubringSubalgebraIntegralClosure {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (A : Subring K) : SMul ↥A ↥(integralClosure (↥A) L)

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

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.