On the p-rank of tame kernel of number fields

Abstract

In this paper, the relations between p-ranks of the tame kernel and the ideal class group for a general number field are investigated. As a result, nearly all of Browkin's results about quadratic fields are generalized to those for general number fields. In particular, a p-rank formula between the tame kernel and the ideal class group for a totally real number field of odd degree is obtained when p is a Fermat prime. As an example, the case of cyclic quartic fields is considered in more details. More precisely, using the results on cyclic quartic fields, we give some results connecting the p-rank(K2OF) with the p-rank(Cl(OE)), where F is a cyclic quartic field and E is an appropriate subfield of F(ζp).