Geodesic
Shafarevich's paper “A general reciprocity law”
Sbornik. Mathematics, Tome 204 (2013) no. 6, pp. 781-800 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A new method for calculating an explicit form of the Hilbert pairing is proposed. It is used to calculate the Hilbert pairing in a classical local field and in a complete higher-dimensional field. Bibliography: 25 titles.
Keywords: explicit reciprocity laws, complete higher-dimensional fields, topological $K$-groups. 
@article{SM_2013_204_6_a0,
     author = {S. V. Vostokov},
     title = {Shafarevich's paper {{\textquotedblleft}A} general reciprocity law{\textquotedblright}},
     journal = {Sbornik. Mathematics},
     pages = {781--800},
     year = {2013},
     volume = {204},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_6_a0/}
}
TY  - JOUR
AU  - S. V. Vostokov
TI  - Shafarevich's paper “A general reciprocity law”
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 781
EP  - 800
VL  - 204
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_6_a0/
LA  - en
ID  - SM_2013_204_6_a0
ER  - 
%0 Journal Article
%A S. V. Vostokov
%T Shafarevich's paper “A general reciprocity law”
%J Sbornik. Mathematics
%D 2013
%P 781-800
%V 204
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2013_204_6_a0/
%G en
%F SM_2013_204_6_a0
S. V. Vostokov. Shafarevich's paper “A general reciprocity law”. Sbornik. Mathematics, Tome 204 (2013) no. 6, pp. 781-800. http://geodesic.mathdoc.fr/item/SM_2013_204_6_a0/

[1] H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz, Erganzungsband, 6, Leipzig, Teubner, 1930 | MR | Zbl

[2] E. Artin, H. Hasse, “Die beiden Ergänzungssätze zum reziprozitätsgesetz der $l^n$-ten potenzreste im körper der $l^n$-ten Einheitswurzeln Einheitswurzeln”, Abh. Math. Sem. Univ. Hamburg, 6:1 (1928), 146–162 | DOI | Zbl

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

[4] S. V. Vostokov, “Explicit form of the law of reciprocity”, Math. USSR-Izv., 13:3 (1979), 557–588 | DOI | MR | Zbl | Zbl

[5] S. V. Vostokov, “Explicit construction of class field theory for a multidimensional local field”, Math. USSR-Izv., 26:2 (1986), 263–287 | DOI | MR | Zbl

[6] H. Brückner, Hilbertsymbole zum Exponenten $p^n$ und Pfaffische Formen, preprint, Hamburg, 1979

[7] H. Hasse, “Die Gruppe der $p^n$-primären Zahlen für einen Primteiler $\mathfrak p$ von $p$”, J. Reine Angew. Math., 176 (1936), 174–183

[8] I. B. Fesenko, S. V. Vostokov, Local fields and their extensions: a constructive approach, Transl. Math. Monogr., 121, Amer. Math. Soc., Providence, RI, 1993 | MR | Zbl

[9] T. B. Belyaeva, S. V. Vostokov, “The Hilbert symbol in a complete multidimensional field. I”, J. Math. Sci. (N. Y.), 120:4 (2004), 1483–1500 | DOI | MR | Zbl

[10] S. V. Vostokov, “On the law of reciprocity in an algebraic number field”, Proc. Steklov Inst. Math., 148 (1980), 75–79 | MR | Zbl

[11] S. V. Vostokov, “Hilbert symbol in a discrete valuated field”, J. Soviet Math., 19:1 (1982), 1006–1019 | DOI | MR | Zbl | Zbl

[12] K. Hensel, “Die multiplikative Darstellungder algebraischen Zahlen für den Bereicheines beliebigen Primteilers”, J. Reine Angew. Math., 136 (1916), 174–183

[13] H. Hasse, Zahlentheorie, Akadenie-Verlag, Berlin, 1963 | MR | Zbl

[14] S. V. Vostokov, I. B. Zhukov, I. B. Fesenko, “On the theory of higher-dimensional local fields. Methods and constructions”, Leningrad Math. J., 2:4 (1991), 775–800 | MR | Zbl | Zbl

[15] A. N. Parshin, “Local class field theory”, Proc. Steklov Inst. Math., 165 (1985), 157–185 | MR | Zbl | Zbl

[16] A. N. Parshin, “On the arithmetic of two-dimensional schemes. I. Distributions and residues”, Math. USSR-Izv., 10:4 (1976), 695–729 | DOI | MR | Zbl | Zbl

[17] S. V. Vostokov, “The pairing on $K$-groups in fields of valuation of rank $n$”, Proceedings of the St. Petersburg Mathematical Society, v. 3, Amer. Math. Soc., Providence, RI, 1995, 111–148 | MR | Zbl

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

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

[20] B. M. Bekker, “Abelian extensions of a complete discrete valuation field of finite heigh”, St. Petersburg Math. J., 3:6 (1992), 1271–1279 | MR | Zbl | Zbl

[21] I. B. Fesenko, “Abelian local $p$-class field theory”, Math. Ann., 301:1 (1995), 561–586 | DOI | MR | Zbl

[22] K. F. Lai, S. V. Vostokov, “The Kneser relation and the Hilbert pairing in multidimensional local field”, Math. Nachr., 280:16 (2007), 1780–1797 | DOI | MR | Zbl

[23] E. Witt, “Schiefkörper über diskret bewerteten Körpern”, J. Reine Angew. Math., 176 (1936), 153–156 | Zbl

[24] S. V. Vostokov, “Hilbert pairing on a complete high-dimensional field”, Proc. Steklov Inst. Math., 208 (1995), 72–83 | MR | Zbl

[25] V. Abrashkin, R. Jenni, “The field-of-norms functor and the Hilbert symbol for higher local fields”, J. Théor. Nombres Bordeaux, 24:1 (2012), 1–39 | DOI | MR | Zbl