Witt rings as integral rings
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) [\textit{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.
scientific article; zbMATH DE number 4023379