Proceedings of the Japan Academy, Series A, Mathematical Sciences

On generic polynomials for the modular 2-groups

Yūichi Rikuna

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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

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

First available in Project Euclid: 23 May 2006

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Primary: 12F12: Inverse Galois theory
Secondary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]

Inverse Galois problem Noether's problem generic polynomials modular 2-groups


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.

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