In theorem A3.5 of Ash's book Abstract Algebra: The Basic Graduate Year (page 20 in this pdf), the author set out to prove the following.
Let $\sigma: F \rightarrow L$ be a field monomorphism where $L$ is algebraically closed. If $E/F$ is algebraic, then $\sigma$ extends to a monomorphism $\tau: E \rightarrow L$.
Here is the full statement and proof:
The proof is essentially an application of Zorn's lemma. However, the author never explicitly explained why it is necessary for $L$ to be algebraically closed.
Here is a counter-example with the condition removed: Stack $E/L/F$ into a chain of extensions where $E$ is algebraically closed and a proper extension of $L$. As a result, we cannot have any monomorphism from $E$ into $L$.
Now, how do I weave this observation into the proof?