An infinite family of pure quartic fields with class number $\equiv 2\pmod{4}$

Abstract

Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$, then the Hilbert class field of $\K$ is $\H_\K=\K(\sqrt{2})$. Finally, we find a criterion to decide if an ideal of the ring of integers or $\K$ is principal or non-principal.