Mathematica Bohemica, Vol. 141, No. 3, pp. 363-384, 2016
On the strongly ambiguous classes of some biquadratic number fields
Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous
Received February 23, 2014. First published June 20, 2016.
Abstract: We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\Bbbk=\Bbb Q(\sqrt{2pq}, i)$, where $ i=\sqrt{-1}$ and $p\equiv-q\equiv1 \pmod4$ 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 $ Am_s(\Bbbk/F)$ and $ Am(\Bbbk/F)$ are also determined from which we deduce that $\Bbbk^{(*)}$ is smaller than the relative genus field $(\Bbbk/\Bbb Q( i))^*$. Then we prove that each strongly ambiguous class of $\Bbbk/\Bbb Q( i)$ capitulates already in $\Bbbk^{(*)}$, which gives an example generalizing a theorem of Furuya (1977).
Keywords: absolute genus field; relative genus field; fundamental system of units; 2-class group; capitulation; quadratic field; biquadratic field; multiquadratic CM-field
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Affiliations: Abdelmalek Azizi, Department of Mathematics, Sciences Faculty, Mohammed First University, Boulevard Mohammed IV, B.P. 524, Oujda, 60000, Morocco, e-mail: abdelmalekazizi@yahoo.fr; Abdelkader Zekhnini, Department of Mathematics, Pluridisciplinary Faculty of Nador, Mohammed First University, B.P. 300, Selouane, Nador, 62700, Morocco, e-mail: zekha1@yahoo.fr; Mohammed Taous, Department of Mathematics, Sciences and Techniques Faculty, Moulay Ismail University, B.P. 509, Boutalamine, Errachidia, 52000, Morocco, e-mail: taousm@hotmail.com
