I need to prove that the ring of invariants $T^G$ of an affine ring $T$ over a field of any characteristic is also affine, in other words, to prove it is finitely generated. This is an exercise Ex 13.3 in GTM150. The hint is that use the remark:
Let $G$ be a finite group acting on a domain $T$, and let $R$ be the ring of invariants, $R = T^G$. Show that every element of $T$ satisfies an integral equation over $R$ - in fact, each element $b\in T$ is integral over the subring generated by the elementary symmetric functions in the conjugates $\sigma b$.
However, I can't see how to use this. If the field $k$ have a characteristic coprime to order of $G$, I know the theorem can be proved use Reynolds operator. But it seems that this does not work here, and it is not what the hint want us to do. I want to see how $R$ is integral over $T$ will help us prove this.