Local class field theory is easy

Abstract

Let K be a local field with finite residue field. For the purposes of this paper, “local class field theory” consists of the (more or less) explicit description of the maximal abelian extension Kab of K, of the calculation of the galois group Gal(K,,/K); i.e., the proof that Gal(KJK) N K*, the completion of K* with respect to the topology given by the open subgroups of finite index in K”, and finally of a description of the isomorphism K* r Gal(K,,/K). L oca c ass field theory in this paper 1 1 does not include, e.g., a calculation of the Brauer group Br(K). It is the aim of this paper, which is partly expository in nature, to show that local class field theory in this sense can be treated briefly and without using any of the involved (but powerful) machinery that one “usually” finds in this connection. In particular we need nothing at all (not even in a concealed way) of the cohomology of groups. All the facts we assume known are collected in Section 2. A large part of this paper (Sections 3,5, 6, and most of 7) is closely related to the authors 1969 Amsterdam thesis. The remaining part of this introduction consists of an outline of the structure of the theory. First let K be a local field with algebraically closed residue field, and let L/K be an abelian (necessarily totally ramified) extension of K. Then one forms the following sequence.