On the 2-rank and 4-rank of the class group of some real pure quartic number fields

TL;DR

This paper investigates the 2-rank and 4-rank of the class group of certain real pure quartic number fields, providing formulas based on prime decomposition and identifying specific forms of the parameter d.

Contribution

It offers explicit calculations of the 2-rank and 4-rank of class groups for a class of real pure quartic fields, linking these ranks to prime decomposition properties.

Findings

Formulas for the 2-rank based on prime divisors of d

Conditions on d for the 2-rank to be 2

Explicit determination of the 4-rank in certain cases

Abstract

Let $K=\mathbb{Q}(\sqrt[4]{pd^{2}})$ be a real pure quartic number field and $k=\mathbb{Q}(\sqrt{p})$ its real quadratic subfield, where $p\equiv 5\pmod 8$ is a prime integer and $d$ an odd square-free integer coprime to $p$. In this work, we calculate $r_2(K)$, the $2$-rank of the class group of $K$, in terms of the number of prime divisors of $d$ that decompose or remain inert in $\mathbb{Q}(\sqrt{p})$, then we will deduce forms of $d$ satisfying $r_2(K)=2$. In the last case, the $4$-rank of the class group of $K$ is given too.