In gtm 167, Field and Galois Theory by Patrick Morandi, to prove the existence of an algebraic closure, the author states that one shall apply Zorn's lemma to find a maximal element once one has proved that for a field $K$ and a set $S$ containing $K$, if $\mathcal{A}$ is the set of all algebraic extension of $K$, then $K\subset L\subset F$ defines a partial order $L\leq F$.
It is easy to verify that the order is well-defined, yet according to Zorn's lemma, to apply it one must firstly find an upper bound for each chain of fields in $\mathcal{A}$, which seems not to be trivial. If $F$ is an upper bound of a chain, then there are no algebraic extension of $K$ except for $K$ itself, this by definition implies that $K$ is an algebraic closure. And this where I got stuck and it definitely doesn't follow the author's original proof. So could you please tell me where the problem is and what is the right way to apply Zorn's lemma?