For a group \(G\) acting linearly on a finite-dimensional vector space \(V\) over a field \(k\), \(G\) acts on the polynomial ring \(k[V]\), and the invariant ring \(k[V]^G\) is the subring of polynomials fixed by \(G\).
Finiteness theorems
Noether. If \(G\) is finite and \(\operatorname{char} k \notdivides \size G\), then \(k[V]^G\) is finitely generated, by invariants of degree at most \(\size G\). The proof uses the Reynolds operator \(f\mapsto {1\over\size G}\sum_{g\in G}g\cdot f\), the averaging used in the proof of Maschke’s theorem.
Hilbert. If \(G\) is linearly reductive, then \(k[V]^G\) is finitely generated; the proof uses the Hilbert basis theorem applied to the ideal of \(k[V]\) generated by the invariants of positive degree.
Computations
\(S_n\) permuting the variables. The invariants are the symmetric polynomials, and \(k[x_1,\ldots,x_n]^{S_n} = k[e_1, \ldots, e_n]\) with \(e_1,\ldots,e_n\) algebraically independent; this is the fundamental theorem of symmetric functions.
\(\ZZ/n\) acting by a root of unity. If a generator acts by \(x_i\mapsto\zeta x_i\) for every \(i\), with \(\zeta\) a primitive \(n\)th root of unity, the invariants are spanned by the monomials of total degree divisible by \(n\).
Molien’s formula. For a finite group \(G\subseteq\GL(V)\) over \(\CC\), \[ \sum_d \dim(\CC[V]^G_d)\, t^d = {1\over \size G}\sum_{g\in G} {1\over \det(1 - tg)}, \] which computes the Hilbert series of the invariant ring, and hence the number of linearly independent invariants in each degree, without finding generators.
The Reynolds operator
If \(\size G\) is invertible in \(k\), the Reynolds operator \(\rho\colon k[V]\to k[V]^G\) is a projection onto \(k[V]^G\) that is \(k[V]^G\)-linear: \(\rho(fh) = f\rho(h)\) for \(f\in k[V]^G\). Hence \(k[V]^G\) is a direct summand of \(k[V]\) as a \(k[V]^G\)-module, which gives finite generation. When \(\operatorname{char}k\) divides \(\size G\), the Reynolds operator is not defined, as in the failure of Maschke's theorem in that case.