The imbedding problem of local fields is considered for the case where the whole of the group is a p-group having as many generators as the Galois group of the extension and the extension consists of a primitive root of 1 of degree equal to the period of the kernel. It is proved that it is necessary and sufficient for the solvability of this problem that a concordance condition (and even a weaker condition) be satisfied (see ).
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S. P. Demushkin and I. P. Shafarevich, “The imbedding problem for local fields,” Izv. Akad. Nauk SSSR, Ser. Matem.,23, No. 6, 823–840 (1959).
S. P. Demushkin, “The group of maximal p-extensions of a local field,” Izv. Akad. Nauk SSSR, Ser. Matem.,25, No. 3, 329–346 (1961).
H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton (1960).
B. B. Lur'e, “On the conditions of imbeddability when the kernel is a nonabelian p-group,” Matem. Zametki,2, No. 3, 233–238 (1967).
Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 91–94, July, 1972.
The author is thankful to A. V. Yakovlev for his valuable advice.
When this article was in press, the author proved that Theorem 1 is valid even without Condition IV. He has also found an example showing that Condition III cannot be discarded.
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Lur'e, B.B. On the imbedding problem for local fields. Mathematical Notes of the Academy of Sciences of the USSR 12, 486–488 (1972). https://doi.org/10.1007/BF01094397
- Local Field
- Weak Condition
- Galois Group
- Primitive Root
- Imbed Problem