Inventxonesmath. 47, 85-99 (1978) Inventione$ matbematicae 9 by Springer-Verlag 1978 On the Structure of Certain Galois Groups Ralph Greenberg* University of Washington, Seattle, WA 98195, USA To Kenkichi Iwasawa 1. Introduction Let k be a number field and let p be a prime. The field obtained by adjoining to k all p-power roots of unity contains a unique subfield k| such that Gal(koo/k)~-Zp, the additive group of p-adic integers. Let Moo denote the maximal abelian pro-p-extension of koo in which only primes of k| lying over p are ramified. One can consider Gal(M| as a A-module in a natural way, where A denotes the complete group ring of Gal(koo/k ) over Z v. (See Section 2.) In [8], Iwasawa proved the following basic result concerning its structure. Theorem. Let r 2 denote the number of complex primes of k. The A-module Gal(M ~/k oo) has rank r 2. It contains no non-trivial finite A-submodule. In this paper, we will generalize Iwasawa's theorem, at least in the case when the ground field k is an abelian extension of Q. We first consider an arbitrary Zfextension K of k. Let G=GaI(K/k) and let A o denote its complete group ring over Zp. Let M r be the maximal abelian pro- p-extension of K unramified outside of the set of primes dividing p and let X K = Gal(MK/K). (We will use this notation if K is any field of algebraic numbers.) Then we will prove the following theorem. Theorem 1. Assume that k is an abelian extension of Q. Then X r has rank r 2 as a Ao-module and it contains no non-trivial finite A6-submodule. In the proof of the above theorem, the only assumption that we actually need about the ground field k is that Leopoldt's conjecture (concerning the p- adic independence of the units of k) is valid. This has been proved by A. Brumer if k is abelian over Q (see [1]). The first conclusion of Theorem 1 (about the rank) follows very simply. Our approach to proving the second conclusion also depends on knowing something about the p-adic independence of units, but in the field k, for all sufficiently large n. (Here kn denotes the subfield of K corresponding to G p" so that k, is a cyclic extension of k of degree pn.) However, the assertion that the rank of X K equals r2 turns out to imply that a certain * This research was supported in part by National ScienceFoundation Grant MCS 7702827