Totally ramified valuations on finite-dimensional division algebras

Abstract

Division algebras D with valuation v are studied, where D is lSnite-dimensional and totally ramified over its center F (i.e., the ramification index of v over wlF equals [D: F]). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When vlF is Henselian a full classification of the F-subalgebras of D is given. When F has a Henselian valuation v with separably closed residue field and A is any tame central simple F-algebra, an algorithm is given for computing the underlying division algebra of A from a suitable subgroup of A*/F*. Some examples are constructed using this valuation theory, including the first example of finitbdimensional F-central division algebras D1 and D2 with D1 XF D2 not a division ring, but D1 and D2 having no common subfield K D F. Valuation theory, long a basic tool in commutative algebra, has been relatively neglected in the study of division algebras, until quite recently. Nontheless, valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties, particularly algebras over iterated Laurent power series fields. For example, such division algebras have been key ingredients in Amitsur's noncrossed product construction [Am] and in Platonov's construction [P] of division algebras D with SK1(D) 7& 1. Valuations are not so prevalent on division algebras as on fields. But if a division algebra D does have a valuation, this structure contains a substantial amount of information about D which would scarcely be accessible otherwise. We consider here valued divisioll algebras D for which D is totQlly ramified and tame over its center F, i.e., for which the ramification index IrD rFI equals the dimension [D: F] of D over F and the characteristic char(D) does not divide [D: F]. (Here rD iS the value group of the valuation on D, and D is the residue division algebra. We assume throughout that [D: F] < oo.) Valued division algebras of this type appear, e.g., in Amitsur's noncrossed product paper [Am, §2], in Saltman's work on indecomposable division algebras [Sa], in certain of the MalcevNeumann division algebras considered by the first author and Amitsur [TA2,§4], etc. However, the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before. This may be because most past work on valued division algebras has concentrated on discrete valuations, when rD -Z; for such a valuation D is never totally ramified over its center F unless D = F (cf. (3.2) below). Received by the editors June 26, 1986. 1980 Mathematics Subject Clmsification (1985 Reon). Primary 16A39. 2Supported in part by F.N.R.S. 2Supported in part by the National Science Foundation. (:)1987 American Mathematlcal Society 0002-9947/87 $}.00 + $ 25 per page