On the unit index of some real biquadratic number fields

Authors: ABDELMALEK AZIZI, ABDELKADER ZEKHNINI, MOHAMMED TAOUS

Abstract: Let $p_1\equiv p_2\equiv1 \pmod 4$ be different prime numbers such that $\left(\dfrac{2}{p_2}\right)=\left(\dfrac{p_1}{p_2}\right)=-\left(\dfrac{2}{p_1}\right)=-1$. Put $\kk=\QQ(\sqrt{2p_1p_2})$ and let $\KK$ be a quadratic extension of $\kk$ contained in its absolute genus field $\kk^{(*)}$. Denote by $k_j$, $1\leq j\leq 3$, the three quadratic subfields of $\KK$. Let $E_{\KK}$ (resp. $E_{k_j}$) be the unit group of $\KK$ (resp. $k_j$). The unit index $\left[E_{\KK}: \prod_{j=1}^3E_{k_j}\right]$ is characterized in terms of biquadratic residue symbols between $2$, $p_1$ and $p_2$ or by the capitulation of $\mathfrak{2}$, the prime ideal of $\QQ(\sqrt{2p_1})$ above $2$, in $\KK$. These results are used to describe the $2$-rank of some CM-fields.

Keywords: Unit index, fundamental systems of units, 2-class group, real biquadratic fields, multiquadratic CM-fields

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