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?
On
I cannot comment on other answers so I will put it here. Let $f=X^q-XY^{q-1}$ and follow the approach in Section 4.6-4.8 of UAC. If $A$ is finite over $B_\alpha=k[X-\alpha Y]=k[X^\ast]$, then $f(X^\ast+\alpha Y, Y)$ as a polynomial in $Y$ has a leading coefficient in $a_m\in k$.
Consider the $f=X^q-XY^{q-1}$ proposed by @user26857 and we find the only term of $Y$ without $X^\ast$ in its coefficients is canceled out as $\alpha^q=\alpha$ for all $\alpha\in k$. Therefore, $Y+f\in A$ is not integral over $k[X-\alpha Y]$ and $A$ is not finite over $B_\alpha$ for all $\alpha\in k$.
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 $(X,Y)\mapsto (X-\alpha Y,Y)$.