## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### On generic polynomials for the modular 2-groups

Yūichi Rikuna

#### Abstract

We construct a generic polynomial for $\mathrm{Mod}_{2^{n+2}}$, the modular 2-group of order $2^{n+2}$, with two parameters over the $2^n$-th cyclotomic field $k$. Our construction is based on an explicit answer for linear Noether's problem. This polynomial, which has a remarkably simple expression, gives every $\mathrm{Mod}_{2^{n+2}}$-extension $L/K$ with $K \supset k$, $\sharp K = \infty$ by specialization of the parameters. Moreover, we derive a new generic polynomial for the cyclic group of order $2^{n+1}$ from our construction.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 3 (2002), 33-35.

Dates
First available in Project Euclid: 23 May 2006

https://projecteuclid.org/euclid.pja/1148392747

Digital Object Identifier
doi:10.3792/pjaa.78.33

Mathematical Reviews number (MathSciNet)
MR1894898

Zentralblatt MATH identifier
1001.12003

#### Citation

Rikuna, Yūichi. On generic polynomials for the modular 2-groups. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 3, 33--35. doi:10.3792/pjaa.78.33. https://projecteuclid.org/euclid.pja/1148392747

#### References

• Hashimoto, K., and Mikake, K.: Inverse Galois problem for dihedral groups. Number Theory and Its Applications (Kyoto, 1997). Dev. Math. vol. 2, Kluwer, Dordrecht, pp. 165–181 (1999).
• Hashimoto, K., and Rikuna, Y.: On generic families of cyclic polynomials with even degree. Manuscripta Math. (To appear).
• Kemper, G.: A constructive approach to Noether's problem. Manuscripta Math., 90, 343–363 (1996).
• Kemper, G., and Mattig, E.: Generic polynomials with few parameters. J. Symbolic Computation, 30, 843–857 (2000).
• Rikuna, Y.: On simple families of cyclic polynomials. Proc. Amer. Math. Soc. (To appear).
• Smith, L.: Polynomial Invariants of Finite Groups. Res. Notes Math. vol. 6, A. K. Peters, Wellesley, MA (1995).