On Tame Kernels and Ideal Class Groups

Abstract

It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the pn-rank of the tame kernel of a cyclic cubic field F with the pn-rank of the coinvariants of \( \mu _{{p^{n} }} \otimes Cl{\left( {{\fancyscript O}_{{E,T}} } \right)} \)under the action of the Galois group, where \( E = F{\left( {\varsigma _{{p^{n} }} } \right)} \) and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F(ζ3).