Generalized Kummer theory and its applications

Komatsu, Toru

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 127-138

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.

Classification : 11R20, 12E10, 12G05
Keywords: Generic polynomial, Kummer theory, Artin symbol

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

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

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

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

[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., Volume 114 (2004), pp. 265-279 | DOI | Zbl | MR

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

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

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

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

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