St. Petersburg Mathematical Journal

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ISSN 1547-7371(e) ISSN 1061-0022(p)

The $\mathbb{Z}_p$ -rank of a topological -group

Author(s): O. Yu. Ivanova
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 569-591.
MSC (2000): Primary 11S70
Posted: June 1, 2009
MathSciNet review: 2473745
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: A complete two-dimensional local field of mixed characteristic with finite second residue field is considered. It is shown that the rank of the quotient $U(1)K_2^{\mathrm{top}}K/T_K$ , where is the closure of the torsion subgroup, is equal to the degree of the constant subfield of over $\mathbb{Q}_p$ . Also, a basis of this quotient is constructed in the case where there exists a standard field containing such that is an unramified extension.

References:

1.: I. B. Fesenko and S. V. Vostokov, Local fields and their extensions. A constructive approach, Transl. Math. Monogr., vol. 121, Amer. Math. Soc., Providence, RI, 1993. MR 1218392 (94d:11095)
2.: S. Lang, Algebraic numbers, Addison-Wesley Publ. Co., Inc., Reading, MA 1964. MR 0160763 (28:3974)
3.: J.-P. Serre, Local fields, Grad. Texts in Math., vol. 67, Springer-Verlag, New York-Berlin, 1979. MR 0554237 (82e:12016)
4.: I. B. Zhukov, Milnor and topological -groups of multidimensional complete fields, Algebra i Analiz 9 (1997), no. 1, 98-147; English transl., St. Petersburg Math. J. 9 (1998), no. 1, 69-105. MR 1458420 (98m:11131)
5.: I. B. Zhukov and A. I. Madunts, Multidimensional complete fields: Topology and other basic constructions, Trudy S.-Peterburg. Mat. Obshch. 3 (1995), 4-46; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 166, Amer. Math. Soc., Providence, RI, 1995, pp. 1-34. MR 1363290 (97b:12009)
6.: I. B. Zhukov and M. V. Koroteev, Elimination of wild ramification, Algebra i Analiz 11 (1999), no. 6, 153-177; English transl., St. Petersburg Math. J. 11 (2000), no. 6, 1063-1083. MR 1746073 (2002a:11134)
7.: I. Zhukov, Higher dimensional local fields, Invitation to Higher Local Fields (Münster, 1999), Geom. Topol. Monogr., vol. 3, Geom. Topol. Publ., Coventry, 2000, pp. 5-18. MR 1804916 (2001k:11245)
8.: O. Hyodo, Wild ramification in the imperfect residue field case, Galois Representations and Arithmetic Algebraic Geometry (Kyoto, 1985/Tokyo, 1986), Adv. Stud. Pure Math., vol. 12, North-Holland, Amsterdam, 1987, pp. 287-314. MR 0948250 (89j:11116)
9.: I. Fesenko, Topological Milnor K-groups of higher local fields, Invitation to Higher Local Fields (Münster, 1999), Geom. Topol. Monogr., vol. 3, Geom. Topol. Publ., Coventry, 2000, pp. 61-74. MR 1804920 (2001k:11247)
10.: -, Theory of local fields. Local class field theory. Multidimensional local class field theory, Algebra i Analiz 4 (1992), no. 3, 1-41; English transl., St. Petersburg Math. J. 4 (1993), no. 3, 403-438. MR 1190770 (93i:11138)
11.: S. Vostokov, Explicit formulas for the Hilbert symbol, Invitation to Higher Local Fields (Münster, 1999), Geom. Topol. Monogr., vol. 3, Geom. Topol. Publ., Coventry, 2000, pp. 81-89. MR 1804922 (2002a:11138)
12.: K. Kato, A generalization of local class field theory by using -groups. I, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 26 (1979), 303-376. MR 0550688 (81b:12016)
13.: O. Yu. Ivanova, Topological -groups of two-dimensional local fields, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 343 (2007), 206-221; English transl., J. Math. Sci. (N. Y.) 147 (2007), no. 5, 7088-1097. MR 2469418

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