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Local class field theory. (English. Russian original) Zbl 1080.11082
St. Petersbg. Math. J. 15, No. 6, 837-846 (2004); translation from Algebra Anal. 15, No. 6, 35-47 (2003).
The author gives condition for a Henselian field $$F$$ which imply that local class field theory is valid for $$F$$ in the sense that for every finite Galois extension $$F_0$$ of $$F$$ the reciprocity map $$\text{Gal}(F_0/F) \to F^{\times}/N_{F_0/F}(F_0^{\times})$$ induces an isomorphism of the projection to $$\text{Gal}(F_0/F)^{ab}$$. He gives an example of a field $$F$$ which shows that these conditions are less restrictive than the applicability of J. Neukirch’s abstract class field theory [Algebraic number theory. Transl. from the German by Norbert Schappacher. Grundlehren der Mathematischen Wissenschaften. 322 (Berlin: Springer) (1999; Zbl 0956.11021), Chapter IV].

##### MSC:
 11S31 Class field theory; $$p$$-adic formal groups
##### Keywords:
Class field theory; Henselian valued field; $$z$$-group
Full Text:
##### References:
 [1] I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, Translations of Mathematical Monographs, vol. 121, American Mathematical Society, Providence, RI, 1993. A constructive approach; With a foreword by I. R. Shafarevich. · Zbl 1156.11046 [2] Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. · Zbl 0956.11021 [3] Yu. L. Ershov, Abstract class field theory (a finitary approach), Mat. Sb. 194 (2003), no. 2, 37 – 60 (Russian, with Russian summary); English transl., Sb. Math. 194 (2003), no. 1-2, 199 – 223. · Zbl 1046.11081 · doi:10.1070/SM2003v194n02ABEH000712 · doi.org [4] -, Multi-valued normed fields, “Nauchn. Kniga,” Novosibirsk, 2000. (Russian) [5] Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. · Zbl 1004.12003 [6] O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. · Zbl 0037.30702 [7] Peter Draxl, Ostrowski’s theorem for Henselian valued skew fields, J. Reine Angew. Math. 354 (1984), 213 – 218. · Zbl 0536.12018 · doi:10.1515/crll.1984.354.213 · doi.org [8] Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. · Zbl 0497.16001
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