# zbMATH — the first resource for mathematics

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

##### MSC:
 1.1e+17 General binary quadratic forms 110000 Quadratic forms over general fields
Full Text:
##### References:
 [1] Knebusch, M., Rosenberg, A., Ware, R.: Structure of Witt rings and quotients of abelian group rings. Am. J. Math.94, 119-155 (1972) · Zbl 0248.13030 · doi:10.2307/2373597 [2] Lam, T.Y.: The Algebraic theory of quadratic forms. Benjamin, Reading Mass. 1973 · Zbl 0259.10019 [3] Leicht, J., Lorenz, F.: Die Primideale des Wittschen Ringes. Invent. Math.10, 82-88 (1970) · Zbl 0227.13015 · doi:10.1007/BF01402972 [4] Pfister, A.: Multiplikative quadratische Formen. Arch. Math.16, 363-370 (1965) · Zbl 0146.26001 · doi:10.1007/BF01220043 [5] Pfister, A.: Quadratische Formen in beliebigen Körpern. Invent. Math.1, 116-132 (1966) · Zbl 0142.27203 · doi:10.1007/BF01389724 [6] Scharlau, W.: Zur Pfisterschen Theorie der quadratischen Formen. Invent. Math.6, 327-328 (1969) · Zbl 0181.04301 · doi:10.1007/BF01425422 [7] Scharlau, W.: Quadratic and hermitian forms. Berlin Heidelberg New York Tokyo: Springer 1985 · Zbl 0584.10010 [8] Witt, E.: Theorie der quadratischen Formen in beliebigen Körpern. J. Reine Angew. Math.176, 31-44 (1937) · Zbl 0015.05701 · doi:10.1515/crll.1937.176.31
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.