$5$-rank of ambiguous class groups of quintic Kummer extensions

Abstract

Let $k=\mathbb{Q}(\sqrt[5]{n},{\zeta }_5)$, where $n$ is a positive integer, $5^{th}$ power-free, whose $5-$class group is isomorphic to $\mathbb{Z}/5\mathbb{Z}\times \mathbb{Z}/5\mathbb{Z}$. Let $k_0=\mathbb{Q}({\zeta }_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity ${\zeta }_5$. Let $C_{k,5}^{(\sigma )}$ the group of the ambiguous classes under the action of $Gal(k/k_0)$ = $<\sigma >$. The aim of this paper is to determine all integers $n$ such that the group of ambiguous classes $C_{k,5}^{(\sigma )}$ has rank $1$ or $2$.

18 pages, 1 figure