On Galois Groups of Abelian Extensions over Maximal Cyclotomic Fields

Let k0 be a finite algebraic number field in a fixed algebraic closure Ω and ζn denote a primitive n-th root of unity ( n ≥ 1). Let k∞ be the maximal cyclotomic extension of k0, i.e. the field obtained by adjoining to k0 all ζn ( n = 1, 2, ...). Let M and L be the maximal abelian extension of k∞ and the maximal unramified abelian extension of k∞ respectively. The Galois groups Gal(M/k∞) and Gal(L/k∞) are, as profinite abelian groups, both isomorphic to the product of countable number of copies of the additive group of Ẑ. Here, Ẑ denotes the profinite completion of the ring of rational integers Z. In fact, more generally, if Msol and Lsol denote the maximal solvable extension of k∞ and the maximal unramified solvable extension of k∞ respectively, the Galois groups Gal(Msol/k∞) and Gal(Lsol/k∞) are both isomorphic to the free prosolvable group on countably infinite generators ( Iwasawa[2], Uchida[5]). On the other hand, as M and L are both Galois extensions of k0, the cyclotomic Galois group Gal(k∞/k0) acts on Gal(M/k∞) and Gal(L/k∞) naturally. The structure of these Galois groups with this action, however, does not seem to be known. Let k1 be the field obtained by adjoining ζ4 and ζp for all odd prime p to k0 and consider the subgroup g = Gal(k∞/k1) of Gal(k∞/k0). It is easy to see that g is isomorphic to the additive group of Ẑ. Now, as Gal(M/k∞) and Gal(L/k∞) are profinite abelian groups, they are naturally Ẑ-modules and g acts on them. Therefore, they can be regarded as A-modules, where A denotes the completed group algebra of g over Ẑ. Our main result is the following