$\mathbb{C}(V)$ is a finite module over $\mathbb{C}(V)^G$?

Question

Is it true that $\mathbb{C}(V)$ is a finite module over $\mathbb{C}(V)^G$ for any finite subgroup $G \subset GL(V)$ and, moreover, $\dim_{\mathbb{C}(V)^G} \mathbb{C}(V) = |G|$? It possibly follows from the well-known theorems of Hilbert and Noether in invariant theory, but doesn't seem obvious to me. Maybe I missed something?

Answer 1

Yes. There is the following result:

Prop Let $V$ be a finite dimensional representation of a finite group $G$ over a field $k$. Then $k(V)$ is galois over $k(V)^G$ with galois group $G$.

The proof is just that $G$ acts as automorphisms on $k(V)$, so we may apply all the standard field theory.

It is immediate from this that your result follows. This result is in Benson's Polynomial Invariants of Finite Groups Prop 1.1.1 (i.e., the very first thing anyone proves in the subject :) )