Field has Algebraic Closure

Theorem
Every field has an algebraic closure.

Proof
Let $F$ be a field.

Let $\mathcal F$ be the collection of all extensions of $F$.

Define an ordering on $\mathcal F$ thus:


 * $\forall K, L \in \mathcal F: K \preceq L \iff L$ is an extension of $K$.

Let $C$ be a chain in $\mathcal F$.

Let $E = \bigcup_{K \mathop \in C} K$.

$E$ satisfies all field axioms, so $E \in \mathcal F$.

By Set is Subset of Union, $E$ is an upper bound for $C$.

By Zorn's Lemma, $C$ has a maximal element $m$.

Then $m$ is an algebraic closure of $F$.