Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 4 / a / ii

For $x\in A$, considerThis is monic, and every $\tau \in G$ permutes its factors, so all coefficients lie in $A^G$. Since $f_x(x)=0$, every $x\in A$ is integral over $A^G$. Thus