Theorems · Theorem · number theory

IsCyclotomicExtension.Rat.mem_intermediateFieldEquivSubgroupChar_iff_conductor_dvd

∀ (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) {m : ℕ} [NeZero m] [IsGalois ℚ ↥F] [IsCyclotomicExtension {m} ℚ ↥F],
  m ∣ n →
    ∀ (χ : DirichletCharacter R n),
      χ ∈ (IsCyclotomicExtension.Rat.intermediateFieldEquivSubgroupChar n K R) F ↔ χ.conductor ∣ m

Assume that m ∣ n, then the image of ℚ(ζₘ) ⊆ ℚ(ζₙ) by intermediateFieldEquivSubgroupChar is the set of characters whose conductor divides m.

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Galois

Cited by

0 results in Mathlib

Foundations

Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NeZeroFieldNumberFieldIsCyclotomicExtensionCommRingHasEnoughRootsOfUnityIsAbelianGaloisNeZeroIsGaloisIsCyclotomicExtension