Received July 27, 2019. Published online October 27, 2020.
Abstract: Let $\Bbbk=\mathbb{Q} \bigl(\sqrt2, \sqrt d \bigr)$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and $\Bbbk_2^{(2)}$ its second Hilbert 2-class field. Denote by $G={\rm Gal}(\Bbbk_2^{(2)}/\Bbbk)$ the Galois group of $\Bbbk_2^{(2)}/\Bbbk$. The purpose of this note is to investigate the Hilbert 2-class field tower of $\Bbbk$ and then deduce the structure of $G$.
Keywords: 2-class group; imaginary biquadratic number field; capitulation; Hilbert 2-class field
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Affiliations: Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Mohammed First University, Mathematics Department, Sciences Faculty, Mohammed V Avenue, P. O. Box 524, Oujda 60000, Morocco, e-mail: 2m.chemseddin@gmail.com, abdelmalekazizi@yahoo.fr; Abdelkader Zekhnini (corresponding author), Mohammed First University, Mathematics Department, Pluridisciplinary Faculty, B.P. 300, Selouane, Nador 62700, Morocco, e-mail: zekha1@yahoo.fr; Idriss Jerrari, Mohammed First University, Mathematics Department, Sciences Faculty, Mohammed V Avenue, P. O. Box 524, Oujda 60000, Morocco, e-mail: idriss_math@hotmail.fr
