Uniform Distribution and Real Fields

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 S(K), undertaken in [6, 71. We maintain the notation of [6, 71. In the first section we investigate U(K) for a real quadratic field K. We calculate generators of U(K) explicitly. In the second section we investigate U(K) for other real fields K. We use a recent result by Yamada to show that 1 U(K) : S(K)/ is infinite for real fields K subject to the restrictions: (1) Q(cJK is cyclic where Q(en) is the least root of unity field containing K. (2) n + 2 (mod 4), and is divisible by at least two distinct primes. We note that a special case of the above result is for K = Q(E~ + .c;‘). In the case where n is odd and divisible by at least two distinct primes then we obtain: S(Q(c, + c;‘)) = S(Q) @o K if and only if n is divisible by a prime congruent to 3 modulo 4. This condition is the exact analog of Yamada’s result on real quadratic fields Q(d1i2) [9, Theorem 7.14, p. 112; 131, viz.: S(Q(d’/“)) = S(Q) @o Q(d1i2) if and only if d is divisible by a prime congruent to 3 modulo 4. In the case where d is not divisible by any prime congruent to 3 modulo 4 then S(K) = U(K)[9,Theorem 7.8, p. 107; 121. As a corollary of the above result we obtain: If (1) K/Q is real of even degree, and (2) the least root of unity field Q(+J containing K has the property that n is odd, divisible by at least 2 distinct primes, and no prime congruent to 1 modulo 4 divides n, then