Uniform Distribution and the Schur Subgroup

In this paper we continue the investigation into the group of algebras with uniformly distributed invariants U(K), and its relation to the Schur subgroup, undertaken in [9]. The notation is the same as in [9]. In the first section we investigate the index 1 U(K), : S(K), 1 where q is an odd prime. We obtain [9, Theorem 2.71 as a special case of Theorem 1.2, wherein we obtain that the above index is infinite when q 1 / K : Q(+)], where +, is the highest q-power root of unity in K, a > 0 provided qa+h +’ /L : K 1, where L is the smallest root of unity field containing K, and ++b is the highest q-power root of unity in L. In the case where aatb 1 / L : K /, the author’s conjecture made in [9] is validated. In Section 2, we show that 1 U(K), : S(K), / is infinite where K/Q is finite, imaginary, and abelian. As an illustration of this result we calculate generators of U(Q(E~,,))~ for primes p + 1 (mod 4) explicitly.