Theorems · Definition · number theory

IsCyclotomicExtension.Rat.subgroupGalEquivSubgroupChar

(n : ℕ) →
  [NeZero n] →
    (K : Type u_1) →
      [inst : Field K] →
        [inst_1 : NumberField K] →
          [hK : IsCyclotomicExtension {n} ℚ K] →
            (R : Type u_2) →
              [inst_2 : CommRing R] →
                [HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)] →
                  Subgroup Gal(K/ℚ) ≃o (Subgroup (DirichletCharacter R n))ᵒᵈ

The bijection between the subgroups of Gal(ℚ(ζₙ)/ℚ) and the subgroups of the group of Dirichlet characters of level n.

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Galois

Cited by

6 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NeZeroFieldNumberFieldIsCyclotomicExtensionCommRingHasEnoughRootsOfUnity