GALOIS GROUPS OF UNRAMIFIED SOLVABLE EXTENSIONS

Throughout this paper, Q, Z and e;n denote the rational numbers, the rational integers and a primitive n-th root of unity for a positive integer n. Let F be an algebraic number field of finite degree. We do not know any general method of determining the structure of the Galois group of the maximal unramified (solvable) extension of F. We mean by "unramified" that every finite or infinite prime is unramified. Let Fn = F(e;n) and let F==UnFn. A. Brumer recently proved that