Isometries of virtual quadratic spaces

Abstract

In this article, we introduce a new object, a virtual quadratic space, and its group of isometries. They are presented as natural generalizations of quadratic spaces and orthogonal groups. It is then shown that by replacing quadratic spaces with virtual quadratic spaces, we can unify certain enumerative properties of finite fields, without distinguishing between even and odd characteristics, such as the number of non-isomorphic non-degenerate quadratic forms, and the order of groups of isometries.