Geodesic
On quadratic subfields of generalized quaternion extensions
Izvestiya. Mathematics , Tome 88 (2024) no. 1, pp. 77-91

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We give necessary and sufficient conditions for the embedding of a quadratic extension of a number field $k$ into an extension with group of generalized quaternions; in this case, the case of both a cyclic kernel and a generalized quaternion is considered. As a consequence, it is proved that the class of ultrasolvable $2$-extensions with cyclic kernel does not coincide with the class of non-semidirect extensions. Sufficient conditions are also given for the embedding of quadratic extensions $k(\sqrt{d_1})/k$, $k(\sqrt{d_2})/k$, $k(\sqrt{d_1d_2})/k$ of a number field $k$ into a generalized quaternion extension $L/k$. Related examples are given.
Keywords: quadratic subfields, ultrasolvability.
Mots-clés : quaternion extensions 
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     title = {On quadratic subfields of generalized quaternion extensions},
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D. D. Kiselev. On quadratic subfields of generalized quaternion extensions. Izvestiya. Mathematics , Tome 88 (2024) no. 1, pp. 77-91. http://geodesic.mathdoc.fr/item/IM2_2024_88_1_a4/