Henselian valued stable fields. (English) Zbl 0919.16010
The author studies relations between the exponent and the index of a finite-dimensional central division algebra over a field \(E\) [cf. e.g. P. M. Cohn, Algebra 3, J. Wiley (1991; Zbl 0719.00002), Ch. 7]. Thus \(E\) is called stable if the index equals the exponent for each such algebra, and \(E\) is called stable closed if all its finite field extensions are stable. The main aim is to characterize stable fields with Henselian valuations. The author showed previously that such a field \(K\) has a residue-class field which is also stable [I. D. Chipchakov, Inst. Math. Bulg. Acad. Sci. No. 1 (1997)]. Here he continues the study of such fields and gives firstly a description of the rank of the Abelian \(p\)-group \(v(K)/p.v(K)\), where \(v(K)\) is the value group and \(p\) any prime, and secondly he describes the cyclic \(p\)-extensions of the residue class field. He also gives a number of sufficient conditions for the stability of Henselian valued fields. He ends by constructing concrete examples of such fields that are stable and also gives a criterion for such fields to be stable closed.
Reviewer: P.M.Cohn (London)
Keywords:
exponent; index; finite-dimensional central division algebras; finite field extensions; stable fields with Henselian valuations; cyclic \(p\)-extensionsCitations:
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