On the Rings of Valuation Vectors

Abstract

Let {R,,,} be a system of topological rings and Qa open subrings of Ra . We consider the set R of all vectors a = (aa), where aa are elements in Ra and which belong to Qa, except for a finite number of a. By the usual definition of component-wise addition and multiplication, R forms a ring containing the direct sum 0 of all Qa . We can then define, in a unique manner, a topology in R so that R becomes a topological ring, 0 becomes an open subring of R and is the Cartesian product of Qa as a topological space. We call R the local direct sum of Ra relative to O. .1 Now, let k be a finite algebraic number field or an algebraic function field of one variable over a constant field F. In the following, we shall call such a field, simply, a number field or a function field respectively. We consider the set {Kp} of all completions of K with respect to prime divisors P of K, which are trivial on F in the case of a function field, and denote by Op the valuation ring of P in Kp, if P is non-archimedean, and the field Kp itself, if P is archimedean. Then, with respect to the usual topology of Kp induced by a valuation of P, each Op is an open subring of Kp , and we can form the local direct sum R of Kp relative to Op . We call this R the ring of valuation vectors of K. If we identify each element E of K with the vector a = (aa), whose components aa are all equal to b, K is isomorphically imbedded in R and becomes a discrete subfield of R, as we shall show later. According to recent results in algebraic number theory, it has become clearer and clearer that the topological properties of R and of its related structures, in particular that of the multiplicative group of R, have essential relations to the arithmetic of the field K.2 Hence, it seems to be of some interest to know what are the characteristic properties of R as a topological ring, for this would show us the sources of arithmetic theorems which can be deduced from the topological structure of R, and might possibly give us some suggestions for further developments in algebraic number theory. The main purpose of the present paper is to give such a characterization of the rings of valuation vectors of number fields and function fields. After some preparations in ?1, we shall do this in ?2 and ?3, and then show in ?4, by some examples, how arithmetic properties of K are related to the topological structure of the ring R. For the proofs, we shall use Haar measures of locally compact groups, some fundamental properties of locally compact rings and, in particular,