Sur l'\'equation $X^2-\varepsilon_2\varepsilon_{p_1p_2}\varepsilon_{2p_1p_2}=0$

TL;DR

This paper investigates the solvability of a specific quadratic equation within a bi-quadratic number field generated by square roots of 2 and the product of two primes congruent to 5 mod 8, focusing on fundamental units.

Contribution

It provides new insights into the solutions of quadratic equations in bi-quadratic fields involving fundamental units of real quadratic subfields.

Findings

Characterization of solutions based on prime congruences

Conditions for the existence of solutions in the field

Relations between fundamental units and solvability

Abstract

Let $p_1\equiv p_2\equiv5 \pmod8$ be prime numbers such that $\left(\frac{p_1}{p_2}\right)=-1$. Let $\mathbb{L}=Q(\sqrt2, \sqrt{p_1p_2})$ Our goal is to resolve the equation $X^2-\varepsilon_2\varepsilon_{p_1p_2}\varepsilon_{2p_1p_2}=0$ in $\mathbb{L}$, where $\varepsilon_j$ are fundamental units of real quadratic subfields of $Q(\sqrt2, \sqrt{p_1p_2})$.