Regular normal extensions over complete fields

Abstract

The local class field theory completely settles all problems pertaining to the abelian extensions of a complete field. All abelian extensions can be described in terms of norm class groups and it can readily be decided which abelian groups can be realized as Galois groups. In this paper we want to attack a more general problem. We consider non-abelian normal extensions of the ground field whose degrees are powers of a prime which is distinct from the characteristic of the residue class field. We construct an infinite normal extension which acts as an universal field. The algebraic approximation of this field yields complete information on the algebraic structure of the field and its Galois group over the ground field. It is shown that the Galois group contains an everywhere dense subgroup which completely suffices to describe the Galois theory. The structure of this group is readily determined. To a certain extent it is a generalization of the fuchsian groups of the classical theory of algebraic functions. The analogy is rather striking. We thus can associate to the given rational prime and the ground field an abstract infinite discrete group which can be considered as a universal covering group with respect to the given prime. All finite groups which can be realized as Galois groups are homomorphic maps of this infinite group. Finally, we associate to certain non-abelian extensions factor groups which are defined in terms of division algebras over the ground field. These factor groups are isomorphic with the respective Galois groups of the fields under consideration. We thus obtain another generalization of local class field theory. In our proofs we make ample use of the ramification theory and local class field theory. Let k be a field which is complete with respect to a discrete valuation of rank one. Suppose that o is the ring of all integers in k and that I= (X) is the prime ideal of o. We shall assume that the residue field o/1 of k is a finite Galois field containing l, = q elements. Let p F 2 be a prime such that q -1 0 (mod p). Then k contains the pth roots of unity.t In the sequel we shall sup-