The Schur subgroup of a real cyclotomic field
Let \(\mathbb Q\) denote the rationals. Let \(n\) be a natural number and \(\zeta_n\) a primitive \(n\)th root of unity. Let \(k\) be a real subfield of the cyclotomic field \(\mathbb Q(\zeta_n)\). The author completely determines the Schur subgroup \(S(k)\) of the Brauer group \(\text{Br}(k)\), if \(\mathbb Q(\zeta_n)/k\) is cyclic. Theorem 1. Let \(n>0\) be an odd integer divisible by at least two distinct primes. Let \(K = \mathbb Q(\zeta_n)\) and \(k\) a real subfield of \(K\) such that \(K/k\) is cyclic. Then \(S(k)\) consists of those classes of \(\text{Br}(k)\) which have uniformly distributed invariants with values 0 or \(\tfrac 12\) such that for a prime number \(p\), the \(p\)-local invariant is 0 whenever \(p\mid n\), and whenever \(p\nmid n\), \(2\mid f_p\), and the 2-part of \(f'_p\) is less than that of \([K:k]\), where \(f_p\) and \(f'_p\) are, respectively, the residue class degrees of \(p\) in \(k/\mathbb Q\) and in \(K/k\). In Theorem 2 of the paper a similar statement is given, when \(n\) is even. The case that \(n\) is a power of a prime was already settled by the author in [J. Algebra 27, 579--589 (1973; Zbl 0277.12009)].
scientific article; zbMATH DE number 3437275