The 3-ranks of tame kernels of cubic cyclic number fields

Abstract

Browkin gave two proofs in [1]. The first proof is analytic and depends on deep results by Mazur and Wiles, while the second one is algebraic, using an exact sequence in K-theory. In this paper, combining the same exact sequence and Gerth’s theory of the 3-class groups of cubic cyclic number fields, we can deal with cubic cyclic number fields with arbitrarily many ramified primes. The main theorem of this paper is Theorem 4.4. From this theorem, one can get the 3-rank formula for general cubic cyclic number fields. As an application, we prove the following theorem in Section 4.