THE 2-RANK OF THE REAL PURE QUARTIC NUMBER FIELD K=ℚ(pd24)

Abstract

We consider the real pure quartic number field $K=\mathbb{Q}(\sqrt[4]{p{d}^2})$, where $p$ is a prime number and $d$ is a square-free positive integer such that $d$ is prime to $p$. We compute $r_2(K)$ the $2$-rank of the class group of $K$ and as an application we exhibit all possible forms of $d$ for which the $2$-class group of $K$ is trivial (equivalently: the class number of $K$ is odd), cyclic or isomorphic to $\mathbb{Z}∕2^{n_1}\mathbb{Z}\times \mathbb{Z}∕2^{n_2}\mathbb{Z}$, where $n_i\in {\mathbb{N}}^{\ast }$.