I was reading Dummit and Foote and their proof of the existence of algebraic closures (Proposition 30 in section 13.4). The proof is from Emil Artin and is Theorem 2.4 in these notes: https://math.hawaii.edu/~rharron/teaching/math612s16/notes.pdf.
I find this proof somewhat ironic considering the books previous statement. In the exposition right before, Dummit and Foote say, "Intuitively, an algebraic closure of $F$ is given by the field 'generated' by all of these fields. The difficulty with this is 'generated' where?, since they are not all subfields of a given field." I feel like the proof that they give in the book succumbs to these same issues. In the book they have an increasing chain of subfields: $K_1 \subseteq K_2 \subseteq K_3 \cdots$. I understand that $K_n$ is identified with an isomorphic copy in $K_{n+1}$ (as $K_{n+1}$ is a quotient of a polynomial ring in $K_n$ by a maximal ideal) and is not a literal subset. However, I cannot make sense of $K = \cup_{n=1}^\infty K_n$. Where is this union taking place?