Remarks on $Z_p$-extensions of number fields

Abstract

Let p be any prime number and k any finite extension of the rational number field Q. Let K be a Zp-extension of k, namely, a Galois extension whose Galois group is topologically isomorphic to the p-adic integer ring For an integer n _ 0, we denote by kn the intermediate field of K/k with degree pn over k and by A, the p-Sylow subgroup of the ideal class group of kn. Let L (resp. Ln) be the p-Hilbert class field of K (resp. kn). Then X G(L/ becomes a module over the formal power series ring A-Z[[] if we define the action of 1 + T by an inner automorphism induced from a fixed topological generator of G(K/k). It is known that X is finitely generated over A. For m a n 0, let wn (1 + D" 1 e A and Pn,m----