On generic polynomials for the modular 2-groups

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.