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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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/

[1] A. V. Yakovlev, “The imbedding problem for fields”, Izv. Akad. Nauk SSSR Ser. Mat., 28:3 (1964), 645–660 (Russian) | MR | Zbl

[2] V. V. Išhanov, “On the semidirect imbedding problem with nilpotent kernel”, Math. USSR-Izv., 10:1 (1976), 1–23 | DOI

[3] V. V. Ishkhanov, B. B. Lur'e, and D. K. Faddeev, The embedding problem in Galois theory, Transl. Math. Monogr., 165, Amer. Math. Soc., Providence, RI, 1997 | DOI | MR | Zbl

[4] D. D. Kiselev, “Minimal $p$-extensions and the embedding problem”, Comm. Algebra, 46:1 (2018), 290–321 | DOI | MR | Zbl

[5] D. D. Kiselev and A. V. Yakovlev, “Ultrasolvable and Sylow extensions with cyclic kernel”, St. Petersburg Math. J., 30:1 (2019), 95–102 | DOI

[6] D. D. Kiselev, “Metacyclic $2$-extensions with cyclic kernel and the ultrasolvability questions”, J. Math. Sci. (N.Y.), 240:4 (2019), 447–458 | DOI

[7] D. D. Kiselev, “Ultrasolvable covering of the group $Z_2$ by the groups $Z_8$, $Z_{16}$ and $Q_8$”, J. Math. Sci. (N.Y.), 219:4 (2016), 523–538 | DOI

[8] D. D. Kiselev, “Ultrasoluble coverings of some nilpotent groups by a cyclic group over number fields and related questions”, Izv. Math., 82:3 (2018), 512–531 | DOI

[9] D. D. Kiselev and B. B. Lur'e, “Ultrasolvability and singularity in the embedding problem”, J. Math. Sci. (N.Y.), 199:3 (2014), 306–312 | DOI

[10] A. V. Jakovlev, “The embedding problem for number fields”, Math. USSR-Izv., 1:2 (1967), 195–208 | DOI | Zbl

[11] G. Malle and B. H. Matzat, Inverse Galois theory, Springer Monogr. Math., Springer-Verlag, Berlin, 1999 | DOI | MR | Zbl

[12] A. V. Yakovlev, “On the embedding problem with cyclic kernel for number fields”, J. Math. Sci. (N.Y.), 192:2 (2013), 243–246 | DOI

[13] V. V. Ishkhanov and B. B. Lur'e, “On the embedding problem with non-Abelian kernel of order $p^4$. V”, J. Math. Sci. (N.Y.), 83:5 (1997), 642–645 | DOI

[14] N. P. Zyapkov and A. V. Yakovlev, “Universally concordant Galois extensions”, J. Soviet Math., 20:6 (1982), 2595–2609 | DOI