Volume 37 • Number 3 • 2014   • The Tame Kernel of Multi-Cyclic Number Fields Xia Wu Abstract. There are many results about the structures of the tame kernels of the number fields. In this paper, we study the structure of those fields $F$, which are the composition of some cyclic number fields, whose degrees over $\mathbb{Q}$ are the same prime number $q$. Then, for any odd prime $p\neq q$, we prove that the $p\mbox{-}$primary part of $K_2\mathcal{O}_F$ is the direct sum of the $p\mbox{-}$primary part of the tame kernels of all the cyclic intermediate fields of $F/\mathbb{Q}$. Moreover, by the approach we developed, we can extend the results to any abelian totally real base field $K$ with trivial $p\mbox{-}$primary tame kernel. 2010 Mathematics Subject Classification: 11R70 Full text: PDF