Generalized Kummer theory and its applications
Komatsu, Toru
Annales mathématiques Blaise Pascal, Tome 16 (2009), p. 127-138 / Harvested from Numdam
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In 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 ζ∉k and ω∈k where ζ is a primitive n-th root of unity and ω=ζ+ζ -1 . In particular, this result with ζ∈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.

Publié le : 2009-01-01
DOI : https://doi.org/10.5802/ambp.259
Classification:  11R20,  12E10,  12G05 
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     author = {Komatsu, Toru},
     title = {Generalized Kummer theory and its applications},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     year = {2009},
     pages = {127-138},
     doi = {10.5802/ambp.259},
     zbl = {1188.11054},
     mrnumber = {2514533},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2009__16_1_127_0}
}

Komatsu, Toru. Generalized Kummer theory and its applications. Annales mathématiques Blaise Pascal, Tome 16 (2009) pp. 127-138. doi : 10.5802/ambp.259. http://gdmltest.u-ga.fr/item/AMBP_2009__16_1_127_0/

[1] Chapman, R. J. Automorphism polynomials in cyclic cubic extensions, J. Number Theory, Tome 61 (1996), pp. 283-291 | Article | MR 1423054 | Zbl 0876.11048

[2] Hashimoto, K.; Rikuna, Y. On generic families of cyclic polynomials with even degree, Manuscripta Math., Tome 107 (2002), pp. 283-288 | Article | MR 1906198 | Zbl 1005.12002

[3] Jensen, C. U.; Ledet, A.; Yui, N. Generic polynomials, Cambridge University Press, Cambridge (2002) | MR 1969648 | Zbl 1042.12001

[4] Kishi, Y. A family of cyclic cubic polynomials whose roots are systems of fundamental units, J. Number Theory, Tome 102 (2003), pp. 90-106 | Article | MR 1994474 | Zbl 1034.11060

[5] Komatsu, T. Potentially generic polynomial (To be submitted)

[6] Komatsu, T. Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory, Manuscripta Math., Tome 114 (2004), pp. 265-279 | Article | MR 2075966 | Zbl 1093.11068

[7] Komatsu, T. Cyclic cubic field with explicit Artin symbols, Tokyo Journal of Mathematics, Tome 30 (2007), pp. 169-178 | Article | MR 2328061 | Zbl 1188.11053 | Zbl pre05202537

[8] Lecacheux, O. Units in number fields and elliptic curves, Advances in number theory, Oxford Univ. Press, New York (1993), pp. 293-301 | MR 1368428 | Zbl 0809.11068

[9] Morton, P. Characterizing cyclic cubic extensions by automorphism polynomials, J. Number Theory, Tome 49 (1994), pp. 183-208 | Article | MR 1305089 | Zbl 0810.12003

[10] Ogawa, H. Quadratic reduction of multiplicative group and its applications, Surikaisekikenkyusho Kokyuroku, Tome 1324 (2003), pp. 217-224 | MR 2000781

[11] Rikuna, Y. On simple families of cyclic polynomials, Proc. Amer. Math. Soc., Tome 130 (2002), pp. 2215-2218 | Article | MR 1896400 | Zbl 0990.12005