On the unit group of some multiquadratic fields

TL;DR

This paper computes the unit groups and 2-class numbers of certain multiquadratic fields generated by square roots of 2 and two primes congruent to 1 mod 4, expanding understanding of their algebraic structure.

Contribution

It provides explicit calculations of units and 2-class numbers for specific multiquadratic fields involving primes congruent to 1 mod 4, which was previously less explored.

Findings

Determined the unit groups of the fields ${K}$ and ${L}$.

Calculated the 2-class numbers of these fields.

Identified conditions affecting the structure of units and class numbers.

Abstract

In this paper, we calculate the unit groups and the $2$-class numbers of the fields $ \mathbb{K}= \mathbb{Q}(\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$ and $ \mathbb{L}= \mathbb{Q}( \sqrt{-1},\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$, where $p_1$ and $p_2$ are two prime numbers satisfying $p_1\equiv p_2 \equiv 1 \pmod{4}$.