Unramified extensions of some cyclic quartic fields

Abstract

In this paper we determine the fourteen unramified extensions for some cyclic quartic fields $K$ whose $2$-class group $C_{K,2}$ is isomorphic to ${\mathbb{Z}}/{2{\mathbb{Z}}}\times {\mathbb{Z}}/{2{\mathbb{Z}}} \times {\mathbb{Z}}/{2{\mathbb{Z}}}$  and to characterize the generators of $C_{K,2}$.