With cedram.org Annales Mathématiques Blaise Pascal Search for an article Search within the site English français Table of contents for this issue | Previous article | Next article Toru KomatsuGeneralized Kummer theory and its applicationsAnnales mathématiques Blaise Pascal, 16 no. 1 (2009), p. 127-138, doi: 10.5802/ambp.259 Article PDF | Reviews MR 2514533 | Zbl 1188.11054 Class. Math.: 11R20, 12E10, 12G05Keywords: Generic polynomial, Kummer theory, Artin symbol Résumé - AbstractIn this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order $n$ and obtain a generalized Kummer theory. It is useful under the condition that $\zeta \notin k$ and $\omega \in k$ where $\zeta$ is a primitive $n$-th root of unity and $\omega =\zeta +\zeta ^{-1}$. In particular, this result with $\zeta \in k$ implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial. Bibliography[1] R. J. Chapman. Automorphism polynomials in cyclic cubic extensions. J. Number Theory, 61:283-291, 1996. Article |  MR 1423054 |  Zbl 0876.11048[2] K. Hashimoto and Y. Rikuna. On generic families of cyclic polynomials with even degree. Manuscripta Math., 107:283-288, 2002. Article |  MR 1906198 |  Zbl 1005.12002[3] C. U. Jensen, A. Ledet and N. Yui. Generic polynomials. Cambridge University Press, 2002  MR 1969648 |  Zbl 1042.12001[4] Y. Kishi. A family of cyclic cubic polynomials whose roots are systems of fundamental units. J. Number Theory, 102:90-106, 2003. Article |  MR 1994474 |  Zbl 1034.11060[5] T. Komatsu. Potentially generic polynomial. To be submitted [6] T. Komatsu. Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory. Manuscripta Math., 114:265-279, 2004. Article |  MR 2075966 |  Zbl 1093.11068[7] T. Komatsu. Cyclic cubic field with explicit Artin symbols. Tokyo Journal of Mathematics, 30:169-178, 2007. Article |  MR 2328061 |  Zbl pre05202537[8] O. Lecacheux, Units in number fields and elliptic curves, Advances in number theory, Oxford Univ. Press, New York, 1993, p. 293-301  MR 1368428 |  Zbl 0809.11068[9] P. Morton. Characterizing cyclic cubic extensions by automorphism polynomials. J. Number Theory, 49:183-208, 1994. Article |  MR 1305089 |  Zbl 0810.12003[10] H. Ogawa. Quadratic reduction of multiplicative group and its applications. Surikaisekikenkyusho Kokyuroku, 1324:217-224, 2003.  MR 2000781[11] Y. Rikuna. On simple families of cyclic polynomials. Proc. Amer. Math. Soc., 130:2215-2218, 2002. Article |  MR 1896400 |  Zbl 0990.12005 © 2019Annales Mathématiques Blaise Pascal Published by the Laboratoire de mathématiques CNRS - UMR 6620 Université Blaise Pascal de Clermont-Ferrand