On Class Numbers of Pure Quartic fields

Let $p$ be a prime. The $2$-primary part of the class group of the pure quartic field $\mathbb{Q}(\sqrt[4]{p})$ has been determined by Parry and Lemmermeyer when $p \not\equiv \pm 1\bmod 16$. In this paper, we improve the known results in the case $p\equiv \pm 1\bmod 16$. In particular, we determine all primes $p$ such that $4$ does not divide the class number of $\mathbb{Q}(\sqrt[4]{p})$. We also conjecture a relation between the class numbers of $\mathbb{Q}(\sqrt[4]{p})$ and $\mathbb{Q}(\sqrt{-2p})$. We show that this conjecture implies a distribution result of the $2$-class numbers of $\mathbb{Q}(\sqrt[4]{p})$.