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 via an institution to check access.
Similar content being viewed by others
Literature cited
J. W. Cassels and A. Frölich (eds.), Algebraic Theory of Numbers, Thompson Book Co., Washington (1967).
B. N. Delone and D. K. Faddeev, “Research on the geometry of Galois theory,” Mat. Sb.,15(57), No. 2, 243–276 (1944).
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).
B. B. Lur'e, “The imbedding problem with noncommutative kernel of order p3,” Tr. Mat. Inst. Akad. Nauk SSSR,80, 98–101 (1965).
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).
A. V. Yakovlev, “The imbedding problem for number fields,” Izv. Akad. Nauk SSSR, Ser. Mat.,31, No. 2, 211–224 (1967).
E. Artin and J. Tate, Class Field Theory, Harvard (1961).
J. Neukirch, “On solvable number fields,” Invent. Math.,53, No. 2, 135–164 (1979).
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
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
Issue Date:
DOI: https://doi.org/10.1007/BF01100115