Symmetric Inner Product Spaces over Z

Abstract

This chapter will discuss the classification problem for inner product spaces over the ring Z of rational integers. All inner products are to be symmetric. Our presentation is based on the classical theorem of Minkowski concerning lattice points in a convex symmetric subset of R ⁿ. This theorem is first used to classify inner product spaces of rank ≦4 over Z. Making use of the Hasse-Minkowski theorem (which we do not prove), it is shown that an indefinite inner product space over Z is completely determined by its rank, type, and signature; where the type is defined to be either I or II according as the space does or does not contain a vector of odd norm. It follows that the Witt ring W(Z) is isomorphic to Z.