On the Structure of the Galois Group of the Maximal Pro-p Extension with Restricted Ramification over the Cyclotomic Z(p)-extension

Let k(infinity) be the cyclotomic Z(p)-extension of an algebraic number field k. We denote by S a finite set of prime numbers which does not contain p, and S(k(infinity)) the set of primes of k(infinity) lying above S. In the present paper, we will study the structure of the Galois group chi(S)(k(infinity)) of the maximal pro-p extension unramified outside S(k(infinity)) over k(infinity). We mainly consider the question whether chi(S)(k(infinity)) is a non-abelian free pro-p group or not. In the former part, we treat the case when k is an imaginary quadratic field and S not equal empty set (here p is an odd prime number which does not split in k). In the latter part, we treat the case when k is a totally real field and S not equal empty set.