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

Abstract

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