# On the rank of the $2$-class group of some imaginary triquadratic number fields

## TL;DR

This paper investigates the rank of the 2-class group in certain imaginary triquadratic fields formed by adjoining an 8th root of unity and a square root of an odd square-free integer.

## Contribution

It provides new insights into the structure of 2-class groups in specific imaginary triquadratic fields, extending understanding of their algebraic properties.

## Key findings

- Determined conditions affecting the 2-class group rank
- Identified patterns in the 2-class group structure for these fields
- Extended existing theories on class groups of number fields

## Abstract

Let $d$ be an odd square-free integer and $\zeta_8$ a primitive $8$-th root of unity. The purpose of this paper is to investigate the rank of the $2$-class group of the fields $L_d=\mathbb{Q}(\zeta_8,\sqrt{d})$.