Geodesic
Schnirelmann's integral and analogy of Cauchy integral theorem for two-dimensional local fields
Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 39-58.

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The problem studied in the thesis arose from the need to find connections between algebraic field theory and theory of functions. The Cauchy integral theorem, which is one of the most basic and classical results of the complex analysis, has a discrete analog in the case of one-dimensional local fields. The natural question then arises whether it is possible to generalize the same result to two-dimensional local fields. The present paper contains the definition of Schnirelmann's integral and the proof of an analog of Cauchy's integral theorem for two-dimensional local fields. As a consequence, links between the Hilbert symbol and Schnirelmann's integral are established.
Keywords: Schnirelmann's integral, analog of Cauchy's integral theorem for two-dimensional local fields. 
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S. V. Vostokov; T. Yu. Shashkov; S. S. Afanas'eva. Schnirelmann's integral and analogy of Cauchy integral theorem for two-dimensional local fields. Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 39-58. http://geodesic.mathdoc.fr/item/CHEB_2020_21_3_a6/

[1] Zhukov I., Invitation to higher local fields, section 1, v. I, Geometry $\$ Topology Monographs, 3, 5–18 | MR

[2] Hasse N., Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, B. G. Teubner, Leipzig, 1930 | MR | Zbl

[3] Kato K., “A generalition of local class field theory by using K-groups. 1”, J. Fac. Sci. Univ. Tokio, Sect. 1 Math., 26:2 (1979), 303–376 | MR | Zbl

[4] Kato K., “A generalization of local class field theory by using K-groups. II”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27:3 (1980), 603–683 | MR | Zbl

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

[6] Vostokov S. V., “Yavnaya konstruktsiya teorii polei klassov mnogomernogo lokalnogo polya”, Izv. AN SSSR. Ser. matem., 49:2 (1985), 283–308 | MR

[7] Vostokov S. V., Zhukov I. B., Fesenko I. B., “K teorii mnogomernykh lokalnykh polei. Metody i konstruktsii”, Algebra i analiz, 2:4 (1990), 91–118 | Zbl

[8] Vostokov S. V., Ivanov M. A., “Integralnaya teorema Koshi i klassicheskii zakon vzaimnosti”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 154, no. 2, 2012, 73–82 | MR

[9] Ivanov M. A., “Proizvedenie simvolov $p^{n}$-kh stepennykh vychetov kak abelev integral”, Algebra i analiz, 24:2 (2012), 120–129 | MR

[10] Lomadze V. G., “K teorii vetvleniya dvumernykh lokalnykh polei”, Matem. sb., 109(151):3(7) (1979), 378–394 | MR | Zbl

[11] Milnor Dzh., Vvedenie v algebraicheskuyu $K$-teoriyu, Mir, M., 1974, 196 pp.

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

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

[14] Fesenko I. B., “Teoriya lokalnykh polei. Lokalnaya teoriya polei klassov. Mnogomernaya lokalnaya teoriya polei klassov”, Algebra i analiz, 4:3 (1992), 1–41

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