Nondegenerate Semiramified Valued and Graded Division Algebras

Degeneracy and decomposability in abelian crossed products

“…In [AS78] the notion of a matrix being degenerate was defined and this notion was further studied and extended in [McK07] and [Mou08]. In this paper we use the original definition given in [AS78].…”

“…Prime power index indecomposable division algebras have been studied in many contexts since a construction of one with ind(D) < exp(D) was given by Saltman in [Sal79]. For example, in light of the present paper, we draw the readers attention to [McK08] and [Mou08] where the decomposability of generic abelian crossed product division algebras are studied. In [Mou08,Cor.…”

“…In [Mou08, Corollary 3.6] it is shown that a p-power degree generic abelian crossed product defined by a non-degenerate matrix (definition given below) is indecomposable. [McK08] and [Mou08] leave open the question of whether any abelian crossed product defined by a non-degenerate matrix is indecomposable. In this paper we answer this question in the negative by constructing a decomposable abelian crossed product F A defined by a non-degenerate matrix (Theorem 2.12/5.2).…”

Kummer subfields of tame division algebras over Henselian fields

“…Remark. We recall that if E is a Henselian valued field and D is an inertially split division algebra over E with D commutative, then D is a tame semiramified division algebra over E (see [M07,Proposition 2.6]). The reader can then see that similar results to Theorem 2.14, Theorem 2.15 in the case of tame semiramified division algebras over a Henselian valued field were proved in [MorSe95].…”

“…Remark 3.3 (1) We recall that we saw in [M07,Proposition 4.6] that if E is a Henselian valued field and D is a nondegenerate tame semiramified division algebra of prime power degree over E, then D has an elementary abelian maximal subfield if and only if Γ D /Γ F is elementary abelian.…”