Algebraically Closed Field is Infinite

Theorem
Algebraically closed fields are infinite.

Proof
We prove the contrapositive: that a finite field cannot be algebraically closed.

Let $F$ be a field, and suppose $F$ is finite.

Since $F$ is finite, there is a polynomial $\displaystyle{f(x) = 1 + \prod_{a\in F} (x-a)}$.

This polynomial cannot have a root in $F$ since for all $a\in F$, $f(a) = 1 + 0 \neq 0$.

Thus, $F$ is not algebraically closed.