Journal of Mathematical Sciences

, Volume 232, Issue 5, pp 662–676 | Cite as

On the Ultrasolvability of p-Extensions of an Abelian Group by a Cyclic Kernel

  • D. D. Kiselev

The paper contains a solution of A. V. Yakovlev’s problem in the embedding theory for p-extensions of odd order with a cyclic normal subgroup and an Abelian quotient group: for such nonsplit extensions there exists a realization for the quotient group as a Galois group over number fields such that the corresponding embedding problem is ultrasolvable (i.e., this embedding problem is solvable and has only fields as solutions). A solution for embedding problems of p-extensions of odd order with kernel of order p and with a quotient group that is represented by a direct product of its proper subgroups is also given – this is a generalization for p > 2 of an analogous result for p = 2 due to A. Ledet.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. V. Ishkhanov, “On a semidirect embedding problem with a nilpotent kernel,” Izv. AN SSSR, Ser. Mat., 40, No. 1, 3–25 (1976).Google Scholar
  2. 2.
    A. V. Yakovlev, “The embedding problem for fields,” Izv. AN SSSR, Ser. Mat., 28, No. 3, 645–660 (1964).zbMATHGoogle Scholar
  3. 3.
    D. D. Kiselev and B. B. Lur’e, “Ultrasolvability and singularity in the embedding problem,” Zap. Nauchn. Semin. POMI, 414, 113–126 (2013).zbMATHGoogle Scholar
  4. 4.
    V. V. Ishkhanov, B. B. Lur’e, and D. K. Faddeev, The Embedding Problem in Galois Theory [in Russian], Nauka, Moscow (1990).Google Scholar
  5. 5.
    D. D. Kiselev, “Examples of embedding problems the solution of which are fields only,” Usp. Mat. Nauk, 68, No. 4, 181–182 (2013).CrossRefGoogle Scholar
  6. 6.
    M. Hall, Group Theory [Russian translation], Izd. Inostr. Liter., Moscow (1962).Google Scholar
  7. 7.
    A. Ledet, “On 2-groups as Galois groups,” Can. J. Math., 47, No. 6, 1253–1273 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. V. Yakovlev, “On ultrasolvable embedding problems for number fields,” Algebra Analiz, 27, No. 6, 260–263 (2015).MathSciNetGoogle Scholar
  9. 9.
    J. W. S. Cassels and A. Fr¨ohlich, Algebraic Number Theory [Russian translation], Mir, Moscow (1969).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Russian Foreign Trade AcademyMoscowRussia

Personalised recommendations