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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Variations on the theme of D. K. Faddeev's paper ``An explicit form of the Kummer-Takagi reciprocity law''


Author: S. V. Vostokov
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
Journal: St. Petersburg Math. J. 19 (2008), 719-722
MSC (2000): Primary 11A15
Published electronically: June 25, 2008
MathSciNet review: 2381941
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Abstract | References | Similar Articles | Additional Information

Abstract: The following form of the Eisenstein reciprocity law is established: in the cyclotomic field $ \mathbb{Q}(\zeta)$, the relation $ (\frac{\alpha}{a})=(\frac{a}{\alpha})$ is equivalent to $ \frac{a^{p-1}-1}{p}\cdot \underline{\alpha}'(1)\equiv 0\mod p$.


References [Enhancements On Off] (What's this?)

  • [E] Gotthold Eisenstein, Mathematische Werke. Band II, Chelsea Publishing Co., New York, 1975. MR 0427030
  • [F] D. K. Faddeev, An explicit form of the Kummer-Takagi reciprocity law, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 1 (1966), 114–122 (Russian). MR 0219513
  • [H] Helmut Hasse, Zahlentheorie, Zweite erweiterte Auflage, Akademie-Verlag, Berlin, 1963 (German). MR 0153659
  • [V] S. V. Vostokov, The reciprocity law in an algebraic number field, Trudy Mat. Inst. Steklov. 148 (1978), 77–81, 273 (Russian). Algebra, number theory and their applications. MR 558942

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Additional Information

S. V. Vostokov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Staryĭ Peterhof, St. Petersburg 198504, Russia
Email: sergeivostokov@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01017-0
Keywords: Reciprocity law, cyclotomic field
Received by editor(s): May 23, 2007
Published electronically: June 25, 2008
Additional Notes: Supported by INTAS
Dedicated: Dedicated to the centenary of the birth of my teacher Dmitriĭ Konstantinovich Faddeev
Article copyright: © Copyright 2008 American Mathematical Society