Free Product Decomposition of Galois Groups of Number Fields

Let G(k(c)|k) be the Galois group of the maximal pro-c extension k(c) of a number field k, where c is a full class of finite groups which is closed under taking subgroups, quotients and extensions. If p is a prime of k and P an extension of p to k(c), then the decomposition group GP(k(c)|k) with respect to P is isomorphic to the Galois group G(kp(c)|kp) of the maximal proc extension kp(c) of kp, cf. [4] theorem (9.3.1). In this paper we consider the question whether the decomposition groups GP(k(c)|k) or the inertia groups TP(k(c)|k) for some primes p form a free pro-c-product inside G(k(c)|k). More precisely, if S(k) and T0(k) are sets of primes of k, then kS(c) is the maximal pro-c extension which is unramified outside S, k0(c) is the maximal pro-c extension which is completely decomposed at T0,