The q-invariant of the Demushkin group G_K(p)

Let K be a nonarchimedean local field containing a primitive pth root of unity, so that the maximal pro-p quotient G_K(p) of its absolute Galois group is a Demushkin group (TauCeti.isDemushkin_absoluteGaloisGroupProP_of_mu). Labute's q-invariant of G_K(p), the order of the torsion subgroup of its topological abelianization (TauCeti.demushkinQ), is the number q(K) of p-power roots of unity in K (TauCeti.localRootOfUnityOrder).

This is the value of q substituted into Labute's marked classification of Demushkin groups when it is applied to G_K(p), alongside the rank [K : ℚ_p] + 2 (TauCeti.demushkinRank_absoluteGaloisGroupProP).

Main results

• TauCeti.demushkinQ_absoluteGaloisGroupProP: q(G_K(p)) = q(K).

References

• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, Theorem 4 and its corollary.
• J.-P. Serre, Structure de certains pro-p-groupes, Séminaire Bourbaki 252 (1962/63).
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (7.5.11).

theorem TauCeti.demushkinQ_absoluteGaloisGroupProP (p : ℕ) [Fact (Nat.Prime p)] (K : Type) [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] (hmu : ∃ (ζ : K), IsPrimitiveRoot ζ p) (h : Finite ↥(pPowerRootsOfUnity p K)) : demushkinQ ⋯ = localRootOfUnityOrder p K h

The q-invariant of G_K(p) is the number of p-power roots of unity of K. For a nonarchimedean local field K containing a primitive pth root of unity, Labute's q-invariant of the Demushkin group G_K(p) is q(K), the order of the group of p-power roots of unity of K.