ℚ(t) and ℚ((t))-Admissibility of Groups of Odd Order

Abstract

Let _Q(t) be the rational function field over the rationals, Q, let Q((t)) be the Laurent series field over Q, and let W be a group of odd order. We investigate the following question: does there exist a finite-dimensional division algebra D central over Q(t) or Q((t)) which is a crossed product for W ? If such a D exists, W is said to be Q(t)-admissible (respectively, Q((t))-admissible). We prove that if W is Q((t))-admissible, then W is also Q(t)-admissible; we also exhibit a Q(t)-admissible group which is not Q((t))admissible. Let K be a field and let ' be a finite group. ' is said to be K-admissible if there exists a division algebra D, finite dimensional and central over K, which is a crossed product for '. Equivalently, ' is K-admissible if there exists a division algebra D with center K having a maximal subfield L Galois over K with Gal(L/K) S '. Admissibility questions for K = Q, the field of rational numbers, have been studied extensively in the literature (e.g., [Sc] and [ SO2 ]). More recently, results have been obtained when K is an algebraic function field over some field Ko ([FSS] and [FS]). In this paper we study admissibility questions for groups of odd order when K is either the rational function field Q(t) or the Laurent series field Q((t)). We show for such groups that Q((t))admissibility implies Q(t)-admissibility but not conversely. We also construct examples of groups of odd order which are Q((t))-admissible but which have homomorphic images which are not Q((t))-admissible; by contrast, if K is a number field, a homomorphic image of a K-admissible group is necessarily K-admissible [Sc, Corollary 2.3]. We fix below most of the basic terminology and notation that we will employ throughout this paper. Let K be a field. By a K-division algebra we mean a division algebra having center K which is finite dimensional over K. We say that A/K is central simple if A is a simple algebra with center K which is finite dimensional over K. Suppose A/K is central simple. By Wedderburn's Theorem, A _ M, (D) where D is a K-division algebra; we refer to D as the division algebra component of A. The Schur index of A, ind(A), equals Received by the editors September 21, 1993. 1991 Mathematics Subject Classification. Primary 12E1 5.