A reciprocity law for maximal fields

Abstract

Introduction. Local class field theory at first was concerned with the abelian extensions of fields which are complete with respect to a non-Archimedean discrete rank one valuation and which have a finite residue class field. Moriya, Nakayama, Schilling [4] and Whaples [6], [9] generalized this theory, allowing the residue class field to be quasifinite, that is, perfect and having for each positive integer n a unique extension of degree n. The condition on the group of values was also relaxed somewhat to include infinite algebraic extensions of such generalized local fields, where the value group need not be infinite cyclic, though it is still of rank one. The purpose of this paper is to show that the reciprocity law holds for a certain class of complete fields with valuations of arbitrary rank. The fields we consider are assumed to be maximal (maximally complete) [3, p. 80], [4, p. 36] with respect to a non-Archimedean valuation. The (multiplicative) value group V will be assumed to satisfy the condition (V: Vn) = n and the residue class field will be essentially quasifinite. Under these conditions we show that there exists a norm residue symbol for abelian extensions which gives the reciprocity law, including the norm transfer and isomorphism transfer properties. The general approach is that of Whaples [1], [6], [9].