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

An element $x\in B$ is integral over $A$ when it satisfies a monic polynomial with coefficients in $A$. Equivalently, $A[x]$ is a finite $A$-module.

If $x,y$ are integral over $A$, then $A[x,y]$ is finite over $A$: it is generated by finitely many monomials $x^i{y}^j$. Multiplication by $x+y$, $x-y$, or $xy$ is an endomorphism of this finite module, so the determinant trick gives a monic annihilating polynomial. Hence the integral elements form a subring $C$ containing $A$.

The integral closure of $A$ in $B$ is this subring $C$. The ring $A$ is integrally closed in $B$ when $C=A$, and $B$ is integral over $A$ when $C=B$.

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