Theorems · Theorem · number theory

IsCyclotomicExtension.Rat.card_intermediateFieldEquivSubgroupChar

∀ (n : ℕ) [inst : NeZero n] (K : Type u_1) [inst_1 : Field K] [inst_2 : NumberField K]
  [hK : IsCyclotomicExtension {n} ℚ K] (R : Type u_2) [inst_3 : CommRing R]
  [inst_4 : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)] [inst_5 : IsAbelianGalois ℚ K]
  (F : IntermediateField ℚ K),
  Nat.card ↥((IsCyclotomicExtension.Rat.intermediateFieldEquivSubgroupChar n K R) F) = Module.finrank ℚ ↥F

The cardinality of the subgroup of Dirichlet characters of level n associated to an intermediate field F of ℚ(ζₙ)/ℚ equals the degree [F : ℚ].

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Galois

Cited by

0 results in Mathlib

Foundations

Depth 219 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NeZeroFieldNumberFieldIsCyclotomicExtensionCommRingHasEnoughRootsOfUnityIsAbelianGalois