On ultrasolvable embedding problems with cyclic kernel. (English) Zbl 1420.12002

Kiselev, D. D.

Russ. Math. Surv. 71, No. 6, 1149-1151 (2016); translation from Usp. Mat. Nauk 71, No. 6, 165-166 (2016).

Continuing studies of Ishkanov, Lur’e and the author himself, the following is proved.
Theorem 1. There is an infinite sries of examples of ultrasolvable embedding problems over number fields with cyclic kernels of order \(n\).
Theorem 2. Let \(k\in\mathbb Q(\varepsilon _n)\) be given. For each prime \(p\mid n\) put \(K_p = k(\sqrt[p^2]{\tilde u_p})\), where \(\tilde u_p\in k^+\) (see Lemma 2). Put \(K =\otimes_{p\mid n}K_p\). Then there exists an ultrasolvable embedding problem of the extension \(K/k\) with cyclic kernel of order \(n\).
The necessary requirements for their proofs are provided in Lemmas 1–3.

Reviewer: Olaf Ninnemann (Uffing am Staffelsee)

MSC:

12F10

Separable extensions, Galois theory