The Noether Map I
Mara D Neusel and M"ufit Sezer

Abstract:
Let $\rho: G\hookrightarrow GL(n, F)$ be a faithful representation of a finite group G.
In this paper we study the image of the associated Noether map 
\[
\eta_G^G: F[V(G)]^G \longrightarrow F[V]^G\/.
\]
It turns out that the image of the Noether map characterizes the ring of 
invariants in the sense that its integral closure $\overline{Im(\eta_G^G)} =F[V]^G$.
This is true without any restrictions on the group, representation, or ground field.
Moreover, we show that the extension $Im (\eta_G^G) \subseteq F[V]^G$ is a finite $p$-root
extension. Furthermore, we show that the Noether map is surjective, i.e., its image 
integrally closed, if $V=F^n$ is a projective $FG$-module. We apply these results and obtain 
upper bounds on the degrees of a  minimal generating set of $F[V]^G$ and the Cohen-Macaulay defect
of $F[V]^G$. We illustrate our results with several examples.