The [wikipedia article](http://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem#Statement_of_the_theorem/%22wikipedia%20article%22) claims that the theorem "was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof". I read the paper by Chevalley and it seems that he only proves the implication: "If the group is generated by pseudo-reflections, then the ring of invariants is polynomial". I wonder whether there is a uniform proof of the inverse implication? Where is it written? 



Chevalley–Shephard–Todd theorem
There are indeed many presentations (if I remember correctly Bourbaki has it)
but the proof is very elegant and short so that I find it hard to refrain from giving
it. Let $H$ be the normal subgroup of the finite $G\subset \mathrm{GL}\_n$
generated by the pseudo-reflections. By the other direction $X:=\mathbb{A}^n/H$
is again affine space and in particular is smooth. We have an action of $G/H$ on
$X$ and a moment's thought reveals that it acts freely in codimension $1$
 (as a point fixed by a non-identity element would lie below a
reflection hyperplane of $\mathbb{A}^n$ and the fixing element below a
pseudo-reflection). Hence $X \to X/(G/H)=\mathbb{A}^n/G$ is étale in codimension
$1$. If $\mathbb{A}^n/G$ were smooth, purity of the branch locus would imply that
the map were étale. However, that forces $G/H$ to act freely on $X$ but the
image of the origin is fixed by all of $G/H$ and therefore $G=H$.


