Advanced Search

Let $K$ be an imaginary cyclic quartic number field whose 2-class group is of type $(2,2,2)$, i.e., isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. The aim of this paper is to determine the structure of the Iwasawa module of the genus field $K^{(*)}$ of $K$.

Let $𝕜=\mathbb{Q}\left(\sqrt{2},\sqrt{d}\right)$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and ${𝕜}_2^{\left(2\right)}$ its second Hilbert $2$-class field. Denote by $G=\mathrm{Gal}({𝕜}_2^{\left(2\right)}/𝕜)$ the Galois group of ${𝕜}_2^{\left(2\right)}/𝕜$. The purpose of this note is to investigate the Hilbert $2$-class field tower of $𝕜$ and then deduce the structure of $G$.