Showing papers on "Generic polynomial published in 1972"

Group rings satisfying a polynomial identity. III

Abstract: Let K[G] denote the group ring of G over the field K and let A denote the F.C. subgroup of G. In this paper we show that if K[G] satisfies a polynomial identity of degree n, then [G: Al] < n/2. Moreover this bound is best possible. If K[G] satisfies a polynomial identity of degree n, then it is known that [G: A] < 0o. In fact if K[G] is prime or if K has characteristic 0 then [G: A] < (n/2)2 by the results of [4]. In general we have [G:/A] < n! by the results of [1]. Thus the goal of this paper is to sharpen these to obtain the best possible bound, namely [G: A] < n/2. We follow the notation of [3]. 1. The abelian case. Throughout this section we assume that [G:/A] < cO and that A is abelian. Let xl = 1, X2, X3, . * Xm be a complete set of m = [G: A] coset representatives for A in G. LEMMA 1. 1. There exists a K-monomorphism p: K[G] -* K[A]m, where the latter is the ring of m x m matrices over K[A], satisfying (i) for a Ec A, p(a) = diag (all, ax2, * ax.), (ii) p(xj)ejj = eil, ellp(xT 1) = eii, where {etj} is the set of matrix units in K[A]m. PROOF. Since A is normal in G, {x11, x"2 1 1, x*-ml} is also a complete set of coset representatives for A in G. Set V = K[G]. Then clearly V is a left K[A]-module with free basis {x7', xi2, , x7m1}. Now V is also a right K[G]-module and as such it is faithful. Since right and left multiplication commute as operators on V, it follows that K[G] is a set of K[A]-linear transformations on an m-dimensional free K[A]-module V. Thus there exists a K-monomorphism p with p(K[G]) c K[A]m. Let a e A. Then x71a = (xT`axj)xT1 = axixT1; so clearly p(a) = diag (axl, a 2, . , axm). Now to compute ellp(xT1) we need only consider the first row of the matrix p(xT'). Since x1x-1 = x7-1 we see that this first row is precisely eli; so ellp(xi-1) = elieli = eli. Received by the editors February 18, 1971. AMS 1970 subject classifications. Primary 16A26; Secondary 16A38.

On the embedding problem for nonsolvable Galois groups of algebraic number fields: Reduction theorems

Abstract: The embedding problem, which is the problem of extending a given Galois extension K 3 k to a Galois extension L 3 K 3 k so that G(L/k) is a prescribed group extension of G(K/k), is investigated in the case k is a number field and G(L/K) is nonsolvable, with respect o the question of reduction methods. Two general (arbitrary k and G(L/K)) reduction theorems are proved, one reducing the general problem to the cases of G(L/K) nilpotent, and split group extensions, resp., and the second reducing the problem in the case G(L/K) having trivial center to the case G(L/k) F aut G(L/K). The notion of localixability of an embedding problem is formulated and investigated for certain classical groups.