Class Fields of Infinite Degree Over p-Adic Number Fields

Abstract

In this paper we shall develop a theory of the infinite abelian algebraic extensions of p-adic number fields. Wetry to characterize these fields by certain groups defined in the ground field. In the case of finite abelian extensions these groups are the norm class groups' and our goal is to find a substitute for these. The natural approach to this is to consider the set of all finite subfields contained in the given field of infinite degree. All these subfields define norm class groups in the ground field. With the help of these groups we are able to define a system of neighbourhoods in the multiplicative group of all elements (not equal to zero) belonging to the ground field. The closure of this topologized group is the generalization of the finite class groups. It is isomorphic to the Galois group of the infinite abelian extension. To establish this fact we use the theorems of the known finite local class field theory. We may mention that the problem is slightly simplified by the remark that the infinite extensions which are considered here are all enumerable. Therefore we need only an enumerable system of neighbourhoods to describe the topology in the space mentioned above. Before going into the details of the construction we first state some well known facts about the local class field theory and the Galois theory of enumerable infinite extensions.