Theorems · Theorem · commutative algebra

IsLocalRing.minpoly_map_residue

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
  [inst_3 : IsLocalRing S] [inst_4 : IsLocalRing R] [inst_5 : Module.Finite R S] [inst_6 : FaithfulSMul R S]
  [Algebra.Etale R S] {β : S},
  R[β] = ⊤ →
    Polynomial.map (IsLocalRing.residue R) (minpoly R β) =
      minpoly (IsLocalRing.ResidueField R) ((IsLocalRing.residue S) β)

For a monogenic étale extension of local rings, the minimal polynomial of β maps to the minimal polynomial of β mod m_S over the residue field.

Defined in

Mathlib.RingTheory.LocalRing.Etale

Cited by

1 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsLocalRingIsLocalRingModule.FiniteFaithfulSMulAlgebra.Etale