Geometric Generalization of Gaussian Period Relations with Application to Noether's Problem for Meta-Cyclic Groups

Abstract

We study Noether's problem over $\mathbb{Q}$ for meta-cyclic groups. This paper is an extension of the previous work [2], which was concerned with the cyclic group $C_n$ of order $n$. We shall give a simple description of the action of the normalizer of $C_n$ in $S_n$ to the function field $\mathbb{Q}(x_1,\dots ,x_n)$, in terms of the generators of the fixed field of $C_n$ given in [2]. Using this, we settle Noether's problem for the dihedral group of order $2n$ $(n\le 6)$ and the Frobenius group of order $20$ with explicit construction of independent generators of the fixed fields. We shall also reconstruct some simple one-parameter families of cyclic and dihedral polynomials.