Witt rings of complete skew fields
The main aim of this paper is to survey Witt rings of skew fields complete relative to a discrete valuation. Witt ring is meant here in the sense of \textit{T. C. Craven} [J. Algebra 77, 74-96 (1982; Zbl 0493.10026)]. We prove that if A is a complete skew field with the residue class field E of characteristic not 2, then the Witt ring W(A) is isomorphic to the group ring \((W(E)/{\mathfrak U}(E_{\sigma}))[\Delta],\) where \(\Delta\) is the two-element group, \({\mathfrak U}(E_{\sigma})\) is the ideal of W(E) generated by all elements of the form \(<1,-d>\), \(d\in \{\sigma (x)x^{-1}:\) \(x\in E^{\times}\}\cdot E^{\times ^ 2}\), and \(\sigma\) is the automorphism of E induced by the inner automorphism \(x\mapsto \pi x\pi ^{-1}\) of A determined by the uniformizer \(\pi\) of A. When A is commutative, then it turns out to be the well known Springer's theorem. The case of dyadic skew fields is also considered. We show that if A is a finite dimensional division algebra over a dyadic field, then every binary form over A is universal.
scientific article; zbMATH DE number 3981258