On the Hilbert $2$-class field tower of some imaginary biquadratic number fields

summary: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 $G$