On the exact divisibility by $5$ of the class number of some pure metacyclic fields

Abstract

Let $\Gamma =\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a natural number $5^{th}$ power-free. Let $k=\mathbb{Q}(\sqrt[5]{n},{\zeta }_5)$, with ${\zeta }_5$ is a primitive $5^{th}$ root of unit, be the normal closure of $\Gamma$, and a pure metacyclic field of degree $20$ over $\mathbb{Q}$. When $n$ takes some particular forms, we show that $\Gamma$ admits a trivial $5$-class group and $5$ divides exactly the class number of $k$.

10 pages, 3 tables