An Application of Pólya's Theory of Counting to an Enumeration Problem Arising in Quadratic Form Theory

The purpose of this paper is to count the number of isomorphism types of reduced Witt rings of nondegenerate symmetric bilinear forms over fields with a fixed number of orderings. All of the background material necessary for understanding the situation in which this counting problem arises will be described in Section 1. Using results from [2], this problem is restated in Section 2 in terms of an equivalent counting problem in graph theory. More specifically, it involves counting all unlabeled rooted trees which have numbers assigned to their vertices satisfying certain conditions. A great deal of work has been done previously by several authors in solving graph enumeration problems [3 1. The techniques we use in Section 2 to solve our counting problem will be entirely combinatorial and can be thought of as an application of these earlier enumeration techniques to a problem arising in quadratic form theory.