Restricted ramification for imaginary quadratic number fields and a multiplicator free group

Abstract

Let K be an imaginary quadratic number field with unit group EK and let t' be a rational prime such that t' IEK . Let S be any finite set of finite primes of K and let K( {, S) denote the maximal e-extension of K (inside a fixed algebraic closure of K) which is nonramified at the finite primes of K outside S. We show that the finitely generated pro-e-group Q(t, S) = Gal(K(t, S)/K) has the property that a complete set of defining relations for Q(t, S) as a pro-{-group can be obtained by lifting the nontrivial abelian or torsion relations in the maximal abelian quotient group U(t, S)ab. In addition we use the key idea of the proof to derive some interesting results on towers of fields over K with restricted ramification. Introduction. If L/K is a Galois extension of fields and E is a Galois extension of K containing L, then E is said to be a central extension of L/K if Gal(E/L) is a subgroup of the centre of Gal(E/K). The study of central extensions of Galois extensions of number fields leads to several arithmetic applications (see [4, Chapter 3]) many of which stem from an early paper of Frohlich [2]. In particular a theorem of Frohlich [4, Theorem 4.11] gives a complete description in terms of topological generators and relations of the Galois group of the maximal tl-extension of Q (t1any prime, Q the field of rational numbers) which is nilpotent of class two with ramification restricted to a finite set of primes. This, in turn, leads to another result of Frohlich [4, Theorem 5.2] which characterizes, in rational terms, all abelian e-extensions L of Q for which edoes not divide the narrow class number of L. Taking the previous discussion as motivation we see that it is important to consider the pro-e-group Q(e, S) = Gal(K(t1, S)/K), where S is a finite set of finite primes of the number field K, and K(t1, S) is the maximal tl-extension of K (inside a fixed algebraic closure of K) which is nonramified at the finite primes of K outside S. The main difficulty in describing Q(t, S) is the determination of a complete set of defining relations for Q(t, S) as a pro-tl-group. When K = Q Frohlich shows that a complete set of defining relations can be obtained by lifting the nontrivial abelian or torsion relations of the maximal abelian quotient group Q(t, S)ab. We explain this idea more precisely in ?1. Our main result shows that Q(e, S) also has the lifting property when K is an imaginary quadratic number field, S is any finite set as above and 1 + I EKI, where EK Received by the editors December 27, 1983 and, in revised form, August 17, 1984. 1980 Mathematics Subject Classification. Primary 12A65. ' This represents part of the author's Ph.D. dissertation written under the direction of Professor Stephen V. Ullom at the University of Illinois at Urbana. ?1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page