Arithmetic of some real triquadratic fields; Units and 2-class groups

TL;DR

This paper computes units and 2-class groups of specific triquadratic fields, identifying families with trivial, cyclic, or Klein group structures for their 2-class groups.

Contribution

It provides explicit computations of units and 2-class groups for certain triquadratic fields, including classification of their 2-class group structures.

Findings

Identified families with trivial 2-class groups

Found fields with cyclic non-trivial 2-class groups

Classified fields with 2-class groups isomorphic to Klein group

Abstract

In this paper, we compute the unit groups and the $2$-class numbers of the Fr\"ohlich's triquadratic fields $\KK=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q})$, where $p$ and $q$ are two prime numbers such that ($p\equiv 1 \pmod8$ and $q\equiv 3 \pmod4$) or ($p\equiv 5$ or $3 \pmod8$ and $q\equiv 3 \pmod4$). Furthermore, we determine some families of the fields $\KK$ whose $2$-class groups are trivial or cyclic non trivial, and some other families with $2$-class groups isomorphic to the Klein group.