Geometric generalization of gaussian period relations with application to noether’s problem for meta-cyclic groups

We study Noether’s problem over Q for meta-cyclic groups. This paper is an extension of the previous work [2], which was concerned with the cyclic group Cn of order n. We shall give a simple description of the action of the normalizer of Cn in Sn to the function field Q(x1,…, xn), in terms of the generators of the fixed field of Cn given in [2]. Using this, we settle Noether’s problem for the dihedral group of order 2n (n ≤ 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.