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> Let $k$ be a field. I have to prove that $\operatorname{Spec}(k[x])$ has infinite points. 
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If $k$ is infinite it is obvious: in fact there are infinite maximal ideals $(x-\alpha\_i)$, with $a\_i \in k$. But if $k$ is a finite field, how can I prove my claim? Is there a general argument like the proof of the infinitude of primes of $\mathbb{Z}$?