Tame kernels of non-abelian Galois extensions of number fields of degree $q^3$

Abstract

Let $E/F$ be a non-abelian Galois extension of number fields of degree $q^{3}$. We give some expressions for the order of the Sylow $p$-subgroup of tame kernel of $E$ and some of its subfields containing $F$, where $p$ is a prime, $q$ is an odd prime, $p\neq q$. As applications, we give some results about the orders of the Sylow $p$-subgroups of tame kernels when $E/mathbb{Q}(\zeta_{3})$ is a Galois extension of number fields with non-abelian Galois group of order $27$.