Theorems · Theorem · number theory

IsCyclotomicExtension.Rat.mem_subgroupGalEquivSubgroupChar_symm_iff

∀ (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)] (σ : Gal(K/ℚ)) (Y : Subgroup (DirichletCharacter R n)),
  σ ∈ (IsCyclotomicExtension.Rat.subgroupGalEquivSubgroupChar n K R).symm (OrderDual.toDual Y) ↔
    ∀ χ ∈ Y, χ ↑((IsCyclotomicExtension.Rat.galEquivZMod n K) σ) = 1

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Galois

Cited by

0 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NeZeroFieldNumberFieldIsCyclotomicExtensionCommRingHasEnoughRootsOfUnity