Let $R = \mathbb{F}_{p^n}[X,Y]/(XY - 1)$.
What are the maximal ideals $M$ of $R$?
What does $R/M$ look like? What is it's degree over $\mathbb{F}_p$?
There is a theorem that states that for every $\mathbb{F}_p$-algebra $R$ of finite type and every maximal ideal $M \subset R$, $R/M$ is a finite field of characteristic $p$.
Now I thought that the ring $R$ above is actually a field isomorphic to $\mathbb{F}_{p^n}(X)$. Since it's a field the only maximal ideal is the trivial one and hence $R/M = R / \{0\} \cong R$. But that is not finite which seems to contradict the theorem. Where did I go wrong?
By the way I haven't yet found a proof of the mentioned theorem.