Calculations of the Schur group

Abstract

Let the field K be an abelian extension of the rational field Q. The Schur group of K, S(K), consists of those classes in the Brauer group of K which contain an algebra isomorphic to a simple component of a rational group algebra QG for some finite group G. Suppose that K has a cyclic extension of the form Q(ζ) where ζ is a primitive nth root of unity. In this paper we calculate the 2-part of S(K) where K contains the fourth roots of unity. An interesting facet of these results is that in some cases certain local indices of classes in S(K) are tied together. That is, a class in S(K) must have a nontrivial local index at an even number of the primes in a certaiji set. The tying together of local indices in these fields is caused by quadratic reciprocity and is not found in the g-part of S(K) where q is an odd prime number. Let [A] be the class in the Brauer group of K which contains the iΓ-central simple algebra A. The Hasse invariant of [A] at a prime © of if is denoted inv® [A]. Benard and Schacher [2] showed that each class [A] in S(K) has uniformly distributed invariants. That is, if the index of [A] is /, and σ(sz) = ej where e7 is a primitive Ith root of unity and σ e Gal (K/Q), then inv© [A] = λ invσ(@) [A] for each prime © in K. A corollary of this result is that the local index of a class [A] in S(K) is the same at each of the primes of K which divide a single rational prime p. This common index is called the p-local index of [A], Set L = Q(ξ) where ξ is a primitive 28nth root of unity, (2, n) = 1.