E. Noether's bound in the invariant theory of finite groups (Q1924586)

Let \(\rho:G\hookrightarrow\text{GL}(n,\mathbb{F})\) be a faithful representation of the finite group \(G\) over the field \(\mathbb{F}\). In 1916, \textit{E. Noether} proved that for \(\mathbb{F}\) of characteristic zero the ring of invariants \(\mathbb{F}[ V]^G\) is generated as an algebra by the invariant polynomials of degree at most \(|G|\). It is known that Noether's bound on the degrees of the generators of \(\mathbb{F}[ V]^G\) holds more generally when the characteristic of \(\mathbb{F}\) is greater than \(|G|\). In this note we prove that Noether's bound holds when \(G\) is solvable and \(|G|\) is relatively prime to the characteristic of \(\mathbb{F}\).