Fields with local class field theory

Abstract

We say that a field k possesses local class jield theory when the finite separable algebraic extensions K / k form an Artin-Tate class formation with respect to the multiplicative groups K *. In addition to the p-adic number fields Q, and the field lR of real numbers themselves, their maximal absolutely algebraic subfields Q; and lRa are known to have local class field theory; in fact, being henselian these latter fields represent the equivalents of Q, , lR for almost all number-theoretic considerations, and they have the additional advantage that they may be commonly embedded in the field a of all algebraic numbers. A close examination shows that not only the finite but also certain precisely distinguishable infinite algebraic extensions k of Qz allow local class field theory. In the present note we show that the fields mentioned here are the only subfields of a with local class field theory. If N j k is a finite or infinite Galois field extension we denote the Galois group of N 1 k by G,,, and in particular we write G, = G,;, when N is the separable algebraic closure k of k. The cohomology groups