# Explicit $H^2(K, \mu) = Q/Z$?

In the development of local class field theory, a very fundamental theorem is that, for every local field $K$ of characteristic zero,

$H^2(K, \mu) \cong \mathbb{Q}/\mathbb{Z}$. $(*)$

Neukirch et al. in Cohomology of Number Fields prove $(*)$ in an indirect way involving the existence of a "dualizing module." I'd like to know if there's an explicit description of the isomorphism $(*)$ (in either direction). In particular, if $L/K$ is a finite extension, then the restriction and corestriction maps $H^2(K, \mu) \to H^2(L, \mu) \to H^2(K, \mu)$ presumably correspond to multiplication by two integers whose product is $[L : K] = n$. What are those integers?

• See Ch. XIII, section 3, Proposition 7 of Serre's Local Fields and the discussion preceding it (and in general read that book for a nice treatment of local class field theory, treating all local fields -- of all characteristics -- on equal footing; e.g. he discusses with proof the concrete meaning in terms of central simple algebras). Serre's article on local class field theory in the book Algebraic Number Theory edited by Cassels and Frohlich is also very illuminating. – nfdc23 Nov 14 '17 at 16:33
• Thank you for the quick answer. Ch. XIII, section 3 indeed contained the material I needed. – Evan O'Dorney Nov 21 '17 at 14:46