Theorems · Theorem · number theory

IsInertiaField.rank_right

∀ (A : Type u_1) (K : Type u_2) (L : Type u_3) {B : Type u_4} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]
  [inst_3 : CommRing A] [inst_4 : CommRing B] [inst_5 : Algebra A B] {p : Ideal A} (P : Ideal B) [P.LiesOver p]
  [FiniteDimensional K L] [inst_8 : MulSemiringAction Gal(L/K) B] [IsGaloisGroup Gal(L/K) A B] [IsDedekindDomain A]
  [IsDedekindDomain B] [Module.Finite A B] [Module.IsTorsionFree A B] [Ring.HasFiniteQuotients A] [P.IsMaximal]
  (E : Type u_6) [inst_16 : Field E] [inst_17 : Algebra E L] [IsInertiaField K L P E] [IsGalois K L]
  [inst_20 : Algebra K E] [IsScalarTower K E L], p ≠ ⊥ → Module.finrank K E = (p.primesOver B).ncard * p.inertiaDegIn B

The degree [E : K] of the inertia field E over K equals the product of the number of prime ideals of B lying over p and the inertia degree of p in B.

Defined in

Mathlib.NumberTheory.RamificationInertia.HilbertTheory

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraCommRingCommRingAlgebraIdeal.LiesOverFiniteDimensionalMulSemiringActionIsGaloisGroupIsDedekindDomainIsDedekindDomainModule.FiniteModule.IsTorsionFreeRing.HasFiniteQuotientsIdeal.IsMaximalFieldAlgebraIsInertiaFieldIsGaloisAlgebraIsScalarTower