root of a unit in a real biquadratic field

Question

Let $p_1$ and $p_2$ two primes numbers $\equiv 1\pmod 4$. If we note by $\varepsilon_m$ the fundamental unit of real quadratic field $\mathbb{Q}(\sqrt m)$, then how can be solved in $\mathbb{Q}(\sqrt{2},\sqrt{p_1p_2})$ the following equation: \begin{equation*} \varepsilon_{2}\varepsilon_{p_1p_2}\varepsilon_{2p_1p_2}=X^2 \end{equation*}

Answer 1

The classical literature on the computation of the unit groups of multiquadratic number fields are

• S. Kuroda, Über den Dirichletschen Körper, J. Fac. Sci. Univ. Tokyo, Sect. I 4 (1943), 383-406 (arithmetic proof of the class number formula)

• S. Kuroda, Über die Klassenzahlen algebraischer Zahlkörper, Nagoya Math. J. 1 (1950), 1-10

• H. Wada, On the class number and the unit group of certain algebraic number fields,
  J. Fac. Sci., Univ. Tokyo, Sect. I 13 (1966), 201-209

The last reference in particular will contain an algorithm for solving your equation.