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Witt rings of complete skew fields. II. (English) Zbl 0638.10013

Wiss. Beitr., Martin-Luther-Univ. Halle Wittenberg 1987/33(M48), 239-243 (1987).
The paper is intended to show an example of a skew field A with the Witt ring W(A) that is not representational. A is chosen as \(K_{\sigma}((t))\), where \(K=F((x))\) and \(\sigma\) is an F-automorphism of K such that \(\sigma (x)=ax\). The result is obtained by combining results of the author’s previous paper [Pac. J. Math. 132, No.2, 391-399 (1988; Zbl 0607.10015)] and the paper of D. B. Shapiro, J.-P. Tignol and A. R. Wadsworth [J. Algebra 78, 58-90 (1982; Zbl 0492.10015)]. The former paper says that \(W(A)\cong (W(F)/<1,a>W(F))[{\mathbb{Z}}_ 2\times {\mathbb{Z}}_ 2],\) whereas the latter allows us to choose F such that \(W(F)/<1,a>W(F)\) is not representational.
Reviewer: A.Sładek

MSC:

11E04 Quadratic forms over general fields
11E16 General binary quadratic forms
12E15 Skew fields, division rings
16Kxx Division rings and semisimple Artin rings