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Ostrowski’s theorem for Henselian valued skew fields. (English) Zbl 0536.12018

Let K be a field having a Henselian valuation v, D a skew field with K contained in its center Z(D) and \(| D:K|<\infty\), e(D/K) the ramification index, \(f(D/K)=| \bar D:\bar K|\) the degree of the extension of residue class fields \=D/\=K. If D is commutative, a theorem of Ostrowski asserts that \(| D:K| =e(D/K) f(D/K) p^ b\) where p is the characteristic of \(\bar K\), and \(b\geq 0\). The main result of this paper extends Ostrowski’s theorem to the case when D is non-commutative. In the course of proving this result the structure of D is analyzed in the special case when \(Z(D)=K\), \(\bar D=\bar K\), and \(p\nmid | D:K|\). It is shown that such a D is isomorphic to a tensor product of cyclic algebras over K.

MSC:

12J20 General valuation theory for fields
12E15 Skew fields, division rings
12J25 Non-Archimedean valued fields
16Kxx Division rings and semisimple Artin rings
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