Algebraically Closed Field is Infinite

Theorem
If $F$ is an algebraically closed field then $F$ is infinite.

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

Suppose that $F$ is finite.

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

Then since a field is a ring, we have by Ring Product with Zero that $\displaystyle{ \prod_{a\in f} (x-a) } = 0$ for all $x \in F$.

Therefore 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.