On the strongly ambiguous classes of some biquadratic number fields

Azizi, Abdelmalek ; Zekhnini, Abdelkader ; Taous, Mohammed

Mathematica Bohemica, Tome 141 (2016) no. 3, pp. 363-384.

Résumé

We study the capitulation of \mbox {$2$-ideal} classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\Bbbk =\Bbb Q(\sqrt {2pq}, {\rm i})$, where ${\rm i}=\sqrt {-1}$ and $p\equiv -q\equiv 1 \pmod 4$ are different primes. For each of the three quadratic extensions $\Bbb K/\Bbbk $ inside the absolute genus field $\Bbbk ^{(*)}$ of $\Bbbk $, we determine a fundamental system of units and then compute the capitulation kernel of $\Bbb K/\Bbbk $. The generators of the groups ${\rm Am}_s(\Bbbk /F)$ and ${\rm Am}(\Bbbk /F)$ are also determined from which we deduce that $\Bbbk ^{(*)}$ is smaller than the relative genus field $(\Bbbk /\Bbb Q({\rm i}))^*$. Then we prove that each strongly ambiguous class of $\Bbbk /\Bbb Q({\rm i})$ capitulates already in $\Bbbk ^{(*)}$, which gives an example generalizing a theorem of Furuya (1977).

Détail

DOI : 10.21136/MB.2016.0022-14

Classification : 11R11, 11R16, 11R20, 11R27, 11R29, 11R37
Keywords: absolute genus field; relative genus field; fundamental system of units; 2-class group; capitulation; quadratic field; biquadratic field; multiquadratic CM-field