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Subfields of nondegenerate tame semiramified division algebras. (English) Zbl 1221.16017

Let \(E\) be a field with a Henselian valuation and \(D\) a nondegenerate central division \(E\)-algebra of degree \(p^n\), for some prime number \(p\) and some \(n\in\mathbb N\). It is well-known that \(D\) has a unique, up-to an equivalence, valuation \(v_D\) extending \(v\). Denote by \(\overline D\) and \(v(D)\) the residue field and the value group of \((D,v_D)\), respectively.
The paper under review is devoted to the study of subfields of \(D\) including \(E\), under the hypothesis that \(D\) is inertially split and semiramified over \(E\). The authors concentrate on the special case where the considered subfields are Galois extensions of \(E\). Assuming that \(K\) is simultaneously a normal extension of \(E\) and a subfield of \(D\), they find several conditions which ensure that \(K/E\) is inertial. For example, this holds when the following two conditions are fulfilled: (i) the quotient group \(v(D)/v(K)\) is noncyclic; (ii) the residue field \(\overline E\) of \((E,v)\) is of characteristic \(p\) or \(K/E\) is a Galois extension, such that the Galois group \(\mathcal G(K/E)\) is elementary Abelian. In the latter case of (ii), this implies that \(D\) is an elementary Abelian crossed product if and only if the Galois group \(\mathcal G(\overline D/\overline E)\) is elementary Abelian. When \(\text{char}(\overline E)=p\), it is proved that \(D\) is a cyclic \(E\)-algebra if and only if \(v(D)/v(E)\) is cyclic. Finally, the paper shows that if \(v(D)/v(E)\) is a group of exponent \(p\) and \(K\) is a subfield of \(D\) including \(E\), then \(K\) is a maximal subfield of \(D\) except, possibly, in the case where \(K/E\) is inertial; when \(K\) is a maximal subfield of \(D\) and \(K/E\) is normal, it proves that \(K/E\) is Galois and determines the structure of \(\mathcal G(K/E)\). – The proofs rely on a careful analysis of the relationship between graded and valued field extensions whose main results are of independent interest.
The authors recall that the study of nondegenerate generic Abelian crossed products along similar lines has been crucial for the proof of the existence of noncrossed product \(p\)-algebras [see D. Saltman, J. Algebra 52, 302-314 (1978; Zbl 0391.13002)]. Their motivation also comes from the result of the first author on the indecomposability of nondegenerate inertially split division algebras of \(p\)-primary degree over Henselian fields, obtained by McKinnie in the case of characteristic \(p\) [see K. McKinnie, J. Algebra 320, No. 5, 1887-1907 (2008; Zbl 1156.16012), and K. Mounirh, Commun. Algebra 36, No. 12, 4386-4406 (2008; Zbl 1167.16014)].

MSC:

16K20 Finite-dimensional division rings
12J10 Valued fields
16K50 Brauer groups (algebraic aspects)
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16W50 Graded rings and modules (associative rings and algebras)
16W70 Filtered associative rings; filtrational and graded techniques
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References:

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