Generalized Kummer theory and its applications

Toru Komatsu[1]

  • [1] Faculty of Mathematics Kyushu University 6-10-1 Hakozaki Higashiku Fukuoka, 812-8581 Japan

Annales mathématiques Blaise Pascal (2009)

  • Volume: 16, Issue: 1, page 127-138
  • ISSN: 1259-1734

Abstract

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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.

How to cite

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Komatsu, Toru. "Generalized Kummer theory and its applications." Annales mathématiques Blaise Pascal 16.1 (2009): 127-138. <http://eudml.org/doc/10565>.

@article{Komatsu2009,
abstract = {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 $\zeta \notin k$ 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.},
affiliation = {Faculty of Mathematics Kyushu University 6-10-1 Hakozaki Higashiku Fukuoka, 812-8581 Japan},
author = {Komatsu, Toru},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Generic polynomial; Kummer theory; Artin symbol},
language = {eng},
month = {1},
number = {1},
pages = {127-138},
publisher = {Annales mathématiques Blaise Pascal},
title = {Generalized Kummer theory and its applications},
url = {http://eudml.org/doc/10565},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Komatsu, Toru
TI - Generalized Kummer theory and its applications
JO - Annales mathématiques Blaise Pascal
DA - 2009/1//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 1
SP - 127
EP - 138
AB - 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 $\zeta \notin k$ 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.
LA - eng
KW - Generic polynomial; Kummer theory; Artin symbol
UR - http://eudml.org/doc/10565
ER -

References

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  1. R. J. Chapman, Automorphism polynomials in cyclic cubic extensions, J. Number Theory 61 (1996), 283-291 Zbl0876.11048MR1423054
  2. K. Hashimoto, Y. Rikuna, On generic families of cyclic polynomials with even degree, Manuscripta Math. 107 (2002), 283-288 Zbl1005.12002MR1906198
  3. C. U. Jensen, A. Ledet, N. Yui, Generic polynomials, (2002), Cambridge University Press, Cambridge Zbl1042.12001MR1969648
  4. Y. Kishi, A family of cyclic cubic polynomials whose roots are systems of fundamental units, J. Number Theory 102 (2003), 90-106 Zbl1034.11060MR1994474
  5. T. Komatsu, Potentially generic polynomial 
  6. T. Komatsu, Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory, Manuscripta Math. 114 (2004), 265-279 Zbl1093.11068MR2075966
  7. T. Komatsu, Cyclic cubic field with explicit Artin symbols, Tokyo Journal of Mathematics 30 (2007), 169-178 Zbl1188.11053MR2328061
  8. O. Lecacheux, Units in number fields and elliptic curves, Advances in number theory (1993), 293-301, Oxford Univ. Press, New York Zbl0809.11068MR1368428
  9. P. Morton, Characterizing cyclic cubic extensions by automorphism polynomials, J. Number Theory 49 (1994), 183-208 Zbl0810.12003MR1305089
  10. H. Ogawa, Quadratic reduction of multiplicative group and its applications, Surikaisekikenkyusho Kokyuroku 1324 (2003), 217-224 MR2000781
  11. Y. Rikuna, On simple families of cyclic polynomials, Proc. Amer. Math. Soc. 130 (2002), 2215-2218 Zbl0990.12005MR1896400

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