Ultrasolvability and Singularity in the Embedding Problem

Kiselev, D. D.; Lur'e, B. B.

doi:10.1007/s10958-014-1858-3

Cited by 7 publications

(2 citation statements)

References 3 publications

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“…It is well known (see Theorem 1 in [9]) that the problem (K/k, G, φ) is ultrasolvable if and only if all its maximal adjoint problems are unsolvable, and, in this case, the problem (K/k, G, φ) itself is solvable. In the case of the quaternion extension (1.2) with n ⩾ 3, the maximal subgroups not containing the kernel ⟨b⟩ have the form G b 2 ,c,n−1 and G b 2 ,cb,n−1 .…”

Section: § 1 Introductionmentioning

confidence: 99%

On quadratic subfields of generalized quaternion extensions

Kiselev

2024

Izv. Math.

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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.

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Section: § 1 Introductionmentioning

confidence: 99%

On quadratic subfields of generalized quaternion extensions

Kiselev

2024

Izv. Math.

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“…We say that the embedding problem (K/k, ϕ 0 ) is adjoined to the initial embedding problem. An embedding problem is said to be ultrasolvable (see [3]) if it has a solution, but all problems adjoined to it are unsolvable. In other words, this means that the problem has a solution, and all of its solutions are fields.…”

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confidence: 99%