Exercise 12, § 2

Let $K$ be a field, $K'$ a finite Galois extension of $K$, $C$ the polynomial ring $K'[X, Y]$, $B$ the quotient ring $C/CXY$ and $x, y$ the canonical images of $X, Y$ in $B$. Let $A'$ be the subring $K[x] + yK'[y]$ of $B$. The Galois group $G$ of $K'$ over $K$ operates faithfully on $A'$, the ring of invariants $A$ of $G$ in $A'$ being $K[x, y]$. Let $S$ be the set of powers of $x$ in $A'$. Show that $S^{-1}A' = S^{-1}A$ and that $G$ does not operate faithfully on $S^{-1}A'$.