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Let $R = \mathbb{F}_{p^n}[X,Y]/(XY - 1)$.

  1. What are the maximal ideals $M$ of $R$?

  2. 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.

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    $R$ is the ring of Laurent polynomials over $\mathbb{F}_{p^n}$. It is not a field since it has many non-invertible elements, such as $1 + X$. You don't need the theorem you cite if you understand the maximal ideals of $\mathbb{F}_{p^n}[X]$ and know some basic things about localization.2012-02-08
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    See if you can show that $R$ is naturally isomorphic to $\mathbb{F}_{p^n}[X]_X$, i.e., the polynomial ring with $X$ inverted. Then use the relationship between prime ideals before and after inverting something to relate these to a particular set of the maximal ideals of $\mathbb{F}_{p^n}[X]$.2012-02-08
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    Thanks for the help. I realised $R$ is isomorphic to the polynomial ring with $X$ inverted, but mistakenly concluded that it was hence a field.2012-02-09

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