Sur l'équation $X^2-{\epsilon }_2{\epsilon }_{p_1{p}_2}{\epsilon }_{2p_1{p}_2}=0$

Abstract

Let $p_1\equiv p_2\equiv 5(\mathrm{mod}8)$ be prime numbers such that $\left(\frac{p_1}{p_2}\right)=-1$. Let $\mathbb{L}=Q(\sqrt{2},\sqrt{p_1{p}_2})$ Our goal is to resolve the equation $X^2-{\epsilon }_2{\epsilon }_{p_1{p}_2}{\epsilon }_{2p_1{p}_2}=0$ in $\mathbb{L}$, where ${\epsilon }_j$ are fundamental units of real quadratic subfields of $Q(\sqrt{2},\sqrt{p_1{p}_2})$.

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