I am reading the book "Algorithms in invariant theory" of Bernd Sturmfels. As Prop 1.1.3/p5, we can see that for every $f\in C{[x]^{{A_n}}}$, $f$ can be written uniquesly in the form $$f = g + h.D$$ where $g,h \in [C{[x]^{{S_n}}}$ and $D = \prod\limits_{1 \leqslant i < j \leqslant n} {({x_i} - {x_j})} $. Thus we can say that $C{[x]^{{A_n}}} = C{[x]^{{S_n}}} \oplus C{[x]^{{S_n}}}$ i.e. $C{[x]^{{A_n}}}$ is a $C{[x]^{{S_n}}}$-free module of rank $[{S_n}:{A_n}]$.
Question: Can we have more general results for any finite groups or a class of finite groups e.g. finite reflection groups,...? This means that for a finite group $G$ and $H \triangleleft G$, then $$C{[x]^H} = {\left( {C{{[x]}^G}} \right)^{[G:H]}}$$
A special another case is $H=1$ and $G$ is a finite reflection group, then $C{[x]^H}=C[x]$ and we have $C{[x]} = {\left( {C[x]^G} \right)^{\left| G \right|}}$
see page 56 in the book "Reflection groups and Coxeter groups" of J. E. Humphreys.
I appreciate if anyone can refer more results to the problem.