Geodesic
Explicit construction of class field theory for a~multidimensional local field
Izvestiya. Mathematics , Tome 26 (1986) no. 2, pp. 263-287.

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Let $k$ be a finite extension of the field of $p$-adic numbers $\mathbf Q_p$ and $k\{\{t\}\}$ the field of Laurent series $\sum_{-\infty}^\infty a_it^i$ for which the $a_i$ are bounded in the norm of $k$ and $a_i\to0$ as $i\to-\infty$. In the $n$-dimensional local field $F=k\{\{t_1\}\}\cdots\{\{t_{n-1}\}\}$ a pairing is given in explicit form between the completed Milnor $k$-functor $K_n^{\mathrm{top}}(F)$ and the multiplicative group $F^*$ with values in the group of $q=p^m$th roots of unity. Bibliography: 14 titles. 
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S. V. Vostokov. Explicit construction of class field theory for a~multidimensional local field. Izvestiya. Mathematics , Tome 26 (1986) no. 2, pp. 263-287. http://geodesic.mathdoc.fr/item/IM2_1986_26_2_a1/

[1] Vostokov S. V., “Yavnaya forma zakona vzaimnosti”, Izv. AN SSSR. Ser. matem., 42:6 (1978), 1288–1321 | MR | Zbl

[2] Vostokov S. V., “Simvol Gilberta v diskretno normirovannom pole”, Zap. nauch. seminarov Leningr. otd. Matem. in-ta AN SSSR, 94, 1979, 50–69 | MR | Zbl

[3] Vostokov S. V., “Simvoly na formalnykh gruppakh”, Izv. AN SSSR. Ser. matem., 45:5 (1981), 985–1014 | MR | Zbl

[4] Ivasava K., Lokalnaya teoriya polei klassov, Mir, M., 1983, 180 pp. | MR

[5] Milnor Dzh., Vvedenie v algebraicheskuyu $K$-teoriyu, Mir, M., 1974, 196 pp. | MR | Zbl

[6] Parshin A. N., “K arifmetike skhem razmernosti 2”, Izv. AN SSSR. Ser. matem., 40:4 (1976), 736–773 | MR | Zbl

[7] Parshin A. N., “Polya klassov i algebraicheskaya $K$-teoriya”, Uspekhi matem. nauk, 30:1 (1975), 253–254 | MR | Zbl

[8] Parshin A. N., “Lokalnaya teoriya polei klassov”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 165, 1984, 143–170 | MR | Zbl

[9] Suslin A. A., “Algebraicheskaya $K$-teoriya i gomomorfizm normennogo vycheta”, Sovr. problemy matem., 25, 1984, 115–208 | MR

[10] Shafarevich I. R., “Obschii zakon vzaimnosti”, Matem. sb., 26(68):1 (1950), 113–146 | Zbl

[11] Bass H., Tate J., “The Milnor ring of a global field”, Lect. Notes Math., 342 (1973), 349–446 | MR | Zbl

[12] Henniart G., “Sur les lois de reciprocite explicites. I”, J. reine und angew. Math., 329 (1981), 177–202 | MR

[13] Hasse H., Zahlentheorie, Berlin, 1963, 611 s | MR

[14] Kato K., “A generalization of local class field theory by using $K$-groups. I”, J. Fac. Sci. Univ. Tokyo, Sect. IA, 26:2 (1979), 303–376 ; “II”, 27:3 (1980), 603–683 | MR | Zbl | MR | Zbl