The Existence of Special Elements

Overview

Let M be a nonsingular quadratic module over a commutative ring R. The primary goal of this chapter is the proof of the following fact: If M is free of finite rank, then C(M) has a special element, or, equivalently, the Arf algebra A(M) of M is isomorphic to R[X](X² - aX - b) for some a and b in R with a² + 4b ∈ R*. This has the important consequence that the results developed in Chapters 7 and 8 for C(M) hold for any free M. In addition, we will establish that the discriminant a² + 4b of X² - aX - b is the classical discriminant of the quadratic module M (except for a factor 2 if rank M is odd). It was established in Chapter 9D that the Arf algebra A(M) is a separable quadratic algebra. The structural features of this algebra will play a decisive role in the proofs. Throughout, R is an arbitrary commutative ring.