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\(K\)-groups of multidimensional local fields. (English. Russian original) Zbl 0708.11065

Ukr. Math. J. 41, No. 2, 237-240 (1989); translation from Ukr. Mat. Zh. 41, No. 2, 266-268 (1989).
It is given a full description of the structure of the group \(K_ 2^{top}(F)\) in the case where \(F\) is a two dimensional local field of mixed characteristics. The proof uses an explicit expression for the Hilbert symbol in the two dimensional case obtained by S. V. Vostokov [Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 2, 283–308 (1985; Zbl 0608.12017)]. As a corollary the author proves that the kernel of the canonical map \(K^ M_ 2(F)\to K_ 2^{top}(F)\) coincides with the infinitely divisible elements in Milnor’s group \(K^ M_ 2(F)\).
Reviewer: A.N.Parshin

MSC:

11S70 \(K\)-theory of local fields
19D45 Higher symbols, Milnor \(K\)-theory

Citations:

Zbl 0608.12017
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References:

[1] S. V. Vostokov, ?An explicit construction of the class field theory of a multidimensional local field,? Izv. Akad. Nauk SSSR, Ser. Mat.,49, No. 2, 283-308 (1985).
[2] A. N. Parshin, ?Class fields and algebraic K-theory,? Usp. Mat. Nauk,30, No. 1, 253-254 (1975). · Zbl 0302.14005
[3] A. N. Parshin, ?Local class field theory,? Trudy, Mat. Inst. Akad. Nauk SSSR,165, 143-170 (1984). · Zbl 0535.12013
[4] I. B. Fesenko, ?The general Hubert symbol in the 2-adic case,? Vestn. Leningr. Univ.,22, 112-114 (1985).
[5] K. Kato, ?A generalization of local class field theory by using K-groups, I,? J. Fac. Sci. Univ. Tokyo, Sect. 1A,26, No. 2, 303-376 (1979). · Zbl 0428.12013
[6] K. Kato, ?A generalization of local class field theory by using K-groups. II,? J. Fac. Sci. Univ. Tokyo, Sect. 1A,27, No. 3, 603-683 (1980). · Zbl 0463.12006
[7] K. Kato, ?The existence theorem for higher local class field theory,? Publ. I.H.E.S.,43, 1-37 (1980).
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