On Capitulation of S-Ideals in Zp-Extensions

Let k be a finite extension of Q and p a prime number. Let K be a Zp-extension of k and S the set of all prime ideals in k which are ramified in K. We denote by A′∞ the p-Sylow subgroup of the S-divisor class group of K. We give a criterion for A′∞=0 which can be applied for general Zp-extensions. Further, we especially investigate the criterion for a totally real number field k in which p splits completely.