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

TL;DR

This paper investigates the divisibility by 5 of class numbers in certain pure metacyclic fields derived from pure quintic fields, identifying specific forms of n where 5 divides the class number of the normal closure.

Contribution

It provides explicit conditions on n for which the class number of the normal closure is divisible by 5, and shows cases where the pure quintic field has a trivial 5-class group.

Findings

5 divides the class number of the normal closure for certain n

Pure quintic fields can have trivial 5-class groups under specific conditions

Explicit forms of n determine 5-divisibility of class numbers

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 $k$.