Field has Algebraic Closure

Theorem
Every field has an algebraic closure.

Proof
Let $F$ be a field and let $\mathcal F$ be the collection of all extensions of $F$. Define a partial order where for all $K, L\in\mathcal{F}$, $K \leq L$ if and only if $L$ is an extension of $K$. Let $C$ be a chain in $\mathcal{F}$. $\bigcup C$ satisfies all field axioms, so $\bigcup C \in \mathcal{F}$ and $\bigcup C$ is an upper bound for $C$. Thus, by Zorn's Lemma, $C$ has a maximal element that is an algebraic closure of $F$.