Do finite algebraically closed fields exist? Let $K$ be an algebraically closed field ($\operatorname{char}K=p$). Denote
$${\mathbb F}_{p^n}=\{x\in K\mid x^{p^n}-x=0\}.$$
It's easy to prove that ${\mathbb F}_{p^n}$ consists of exactly $p^n$ elements.
But if $|K|<p^n$, we have collision with previous statement (because ${\mathbb F}_{p^n}$ is subfield of $K$).
So, are there any finite algebraically closed fields? And if they exist, where have I made a mistake?
Thanks.

Hint $\rm\:F[x]\:$ has infinitely many primes for every field $\rm\,F\,$ -- by mimicking Euclid's proof for integers. In particular, if $\rm\,F\,$ is  algebraically closed, there are infinitely many nonassociate primes $\rm\ x - a_i\:$ therefore there are infinitely many elements $\rm\:a_i\in F\:.\:$
Remark $\ $ This explains the genesis of the polynomial employed in Zev's answer. 
