On Class Numbers of Pure Quartic fields

TL;DR

This paper advances understanding of the class numbers of pure quartic fields, especially for primes congruent to ±1 mod 16, and proposes a conjecture linking class numbers of related fields.

Contribution

It improves known results on the 2-primary part of class groups for primes p ≡ ±1 mod 16 and introduces a conjecture relating class numbers of different quadratic fields.

Findings

Determined all primes p where 4 does not divide the class number of Q(∛p)

Extended results to primes p ≡ ±1 mod 16

Proposed a conjecture linking class numbers of Q(∛p) and Q(√-2p)

Abstract

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 $\mathbb{Q}(\sqrt[4]{p})$.