Noether's problem for some meta-abelian groups of small degree

Abstract

In this note we solve Noether's problem over $\mathbf{Q}$ for some meta-abelian groups of small degree $n$. Let $G$ be a subgroup of the group of one-dimensional affine transformations on $\mathbf{Z}/n\mathbf{Z}$ which contains $\mathbf{Z}/n\mathbf{Z}$. For $n=9,10,12,14,15$, we show that Noether's problem for $G$ has an affirmative answer by constructing an explicit transcendental basis of the fixed field over $\mathbf{Q}$.