Witt rings as integral rings.(English)Zbl 0629.10017

Let F denote a field of characteristic not two, and W(F) the Witt ring of classes of nondegenerate symmetric bilinear forms over F. It has been known since the definition of W(F) [E. Witt, J. Reine Angew. Math. 176, 31-44 (1936; Zbl 0015.05701)] that W(F) is an integral extension of $${\mathbb{Z}}$$. In this paper the author finds explicit monic polynomials in $${\mathbb{Z}}[x]$$ that annihilate the classes of all forms of a fixed dimension: Thus, for n even, let $$p_ n(x)=x(x^ 2-2^ 2)(x^ 2-4^ 2)...(x^ 2-n^ 2)$$ for n odd, let $$p_ n(x)=(x^ 2-1^ 2)(x^ 2- 3^ 2)...(x^ 2-n^ 2).$$ Then, if f denotes a nondegenerate symmetric bilinear form of dimension n, and $$\bar f$$ its class in W(F), then $$p_ n(\bar f)=0$$. Other polynomials, $$t_ n$$, in $${\mathbb{Z}}[x]$$, are also given, with $$t_ n(\bar f)=0$$, if f has determinant 1. Finally, if $$\bar f$$ is in $$I^ n$$, where I denotes the so-called fundamental ideal of even dimensional forms, a polynomial in $${\mathbb{Z}}[x]$$ annihilating $$\bar f$$ is determined.
Reviewer: A.F.T.W.Rosenberg