$k[X,Y]/(f)$ not finitely generated as a module (Exercise 4.10 Reid, UCA) I have been wrestling with this problem for some time and I still can't find $f$. It seems really simple, which annoys me even more. The problem is as follows (Exercise 4.10 Reid, UCA):

Suppose $k= \mathbb{F}_q$ is a finite field with $q$ elements;  give an exmaple of $f \in k[X,Y]$ such that for every $\alpha \in k$, the ring $A=k[X,Y]/(f)$ is not finitely generated as a module over $B_{\alpha}=k[X-\alpha Y]$.

It's the fact that $A$ is not finitely generated that bothers me. Any help?

Hint. Try $f=X^q-XY^{q-1}$. See what's going on for $\alpha=0$, and extend this to any $\alpha$ by changing the variables $(X,Y)\mapsto (X-\alpha Y,Y)$.
