Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Imbedding problem with non-Abelian kernel of order p4

  • 14 Accesses

Abstract

This paper considers the imbedding problem for numerical fields and p-groups with nonabelian kernel of order p4, two generators α and β, and defining relations\(\alpha ^{p^2 } \)=1, βp=1, [α,β,α]=1, and [α,β,β]= 1. For p=2 and “almost always” for odd p, the Hasse principle is valid,” and the problem is solvable if and only if all related local problems are solvable. Counterexamples in which the Hasse principle is not valid are constructed for some exceptional cases.

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    J. W. Cassels and A. Frölich (eds.), Algebraic Theory of Numbers, Thompson Book Co., Washington (1967).

  2. 2.

    B. N. Delone and D. K. Faddeev, “Research on the geometry of Galois theory,” Mat. Sb.,15(57), No. 2, 243–276 (1944).

  3. 3.

    S. P. Demushkin and I. R. Shafarevich, “The imbedding problem for local fields,” izv. Akad. Nauk SSSR, Ser. Mat.,23, No. 6, 823–840 (1959).

  4. 4.

    B. B. Lur'e, “The imbedding problem with noncommutative kernel of order p3,” Tr. Mat. Inst. Akad. Nauk SSSR,80, 98–101 (1965).

  5. 5.

    B. B. Lur'e, “The compatibility condition in the field imbedding problem,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad Nauk SSSR,71, 155–162 (1977).

  6. 6.

    A. V. Yakovlev, “The imbedding problem for number fields,” Izv. Akad. Nauk SSSR, Ser. Mat.,31, No. 2, 211–224 (1967).

  7. 7.

    E. Artin and J. Tate, Class Field Theory, Harvard (1961).

  8. 8.

    J. Neukirch, “On solvable number fields,” Invent. Math.,53, No. 2, 135–164 (1979).

Download references

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 46–62, 1989.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ishkhanov, V.V., Lur'e, B.B. Imbedding problem with non-Abelian kernel of order p4 . J Math Sci 57, 3473–3481 (1991). https://doi.org/10.1007/BF01100115

Download citation

Keywords

  • Local Problem
  • Exceptional Case
  • Numerical Field
  • Hasse Principle