The Schur Group of an Algebraic Number Field

Abstract

If G denotes a finite group and Q the rational field, then the group algebra Q[G] is a direct sum of simple rings, each a full matrix ring over a division ring. One may ask which simple rings arise this way as G is allowed to range over all finite groups. The center of such a simple ring is a field K which is a subfield of Q(sm), where sm is a primitive mth root of unity and m divides the order of G. If we fix our attention on one particular field K, the subset of the Brauer group B(K) of K consisting of all those classes containing an algebra which is isomorphic to a simple summand of Q[G] for some G is a subgroup, S(K), the Schur group of K. This is the same group as the set of classes in B(K) which contain a simple summand of K[G] for some G. We take this as the definition of S(K) when K is any field. Since the elements of B(K) are uniquely determined by Hasse invariants when K is an algebraic number field, it is natural to ask for a description of S(K) in these terms. The main purpose of this paper is to give a description of S(K)p, the p-primary subgroup of S(K), in the case p is odd and K is any abelian extension of Q. In other words we describe up to equivalence in the Brauer group, those simple algebras of odd index which can appear as a direct summand of Q[G]. Some information about the 2-primary part of S(K) is also obtained in the case K contains s, a fourth root of unity. For such a field we determine which numbers can occur as a local index of an element in S(K). Let Q(sm) = L be the least root of unity field containing the given K and let R be the set of rational primes dividing m which ramify between K and L with ramification index divisible by the prime p. Let S(K, R)p denote the subgroup of S(K)p consisting of all elements split at all primes of K which do not divide an element of A. For a prime q not dividing m, let S(K, q)p denote the subgroup of S(K)p of elements split everywhere except at the divisors of q. We obtain the following result.