Theorems · Theorem · number theory

Ideal.card_inertia_eq_ramificationIdxIn

∀ {R : Type u_1} {S : Type u_2} {G : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
  [inst_3 : Group G] [inst_4 : MulSemiringAction G S] [IsGaloisGroup G R S] [Finite G] [IsDomain R] [IsDomain S]
  [Module.Finite R S] [Module.Flat R S] (p : Ideal R) (P : Ideal S) [P.LiesOver p] [inst_12 : p.IsPrime] [P.IsPrime]
  [PerfectField p.ResidueField], Nat.card ↥(Ideal.inertia G P) = p.ramificationIdxIn S

The cardinality of the inertia group is equal to the ramification index.

Defined in

Mathlib.NumberTheory.RamificationInertia.Galois

Cited by

2 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraGroupMulSemiringActionIsGaloisGroupFiniteIsDomainIsDomainModule.FiniteModule.FlatIdeal.LiesOverIdeal.IsPrimeIdeal.IsPrimePerfectField