TITLE: $\mathbb C[X_1, \ldots, X_n]$ is a free module over $\mathbb C[X_1, \ldots, X_n]^G$
QUESTION [6 upvotes]: Let $G$ be finite subgroup of $GL_n( \mathbb C )$. Let $\mathbb C[X_1, \ldots, X_n]^G$ be the set of all   G-invariant polynomials of $\mathbb C[X_1, \ldots, X_n]$. Is there any rule by which we can find out when $\mathbb C[X_1, \ldots, X_n]$ is a free module over $\mathbb C[X_1, \ldots, X_n]^G$?

REPLY [6 votes]: An element of $G$ is called a $\textit{pseudoreflection}$ if it fixes a codimension one subspace of $V:= \mathbb{C}^n$. Let us denote $\mathbb{C}[X_1, \ldots, X_n] = \mathbb{C}[V]$. Then as alluded to in the comments, we have the following: 
$\textbf{Theorem (Chevalley-Shephard-Todd)}$ The following are equivalent: 

$G$ is generated by pseudoreflections;
$\mathbb{C}[V]^G$ is isomorphic to a polynomial ring;
$\mathrm{Spec} (\mathbb{C}[V]^G)$ is nonsingular;
$\mathbb{C}[V]$ is a free module over $\mathbb{C}[V]^G$.