Geodesic
On the Hilbert $2$-class field tower of some imaginary biquadratic number fields
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 1, pp. 269-281
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Let $\Bbbk =\mathbb {Q} \bigl (\sqrt 2, \sqrt d \bigr )$ 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 $G$.
Let $\Bbbk =\mathbb {Q} \bigl (\sqrt 2, \sqrt d \bigr )$ 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 $G$.
DOI : 10.21136/CMJ.2020.0333-19
Classification : 11R11, 11R27, 11R29, 11R37
Keywords: $2$-class group; imaginary biquadratic number field; capitulation; Hilbert $2$-class field 
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Chems-Eddin, Mohamed Mahmoud; Azizi, Abdelmalek; Zekhnini, Abdelkader; Jerrari, Idriss. On the Hilbert $2$-class field tower of some imaginary biquadratic number fields. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 1, pp. 269-281. doi: 10.21136/CMJ.2020.0333-19

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