When is the algebraic closure a colimit? Let $k$ be a field. We know that there exists an algebraically closed field $\bar{k}$ containing $k$ such that $\bar{k}/k$ is algebraic. This is the algebraic closure of $k$.
Usually this is constructed as the quotient of a huge polynomial ring. Nevertheless, I know that in some cases we can write $\bar{k}$ as the (directed) colimit of all finite subextensions of $\bar{k}/k$. (This is a colimit in the category of extensions of $k$. Not a colimit in the category of fields, I think.) This should hold for finite fields, if I recall correctly.
I now wonder two things:

*

*Under which conditions on $k$ does this result holds?

*If it holds, does it means that $\bar{k}$ is equal to the union of every finite subextension of $k$?

 A: For any field $K$ and any directed family of subfields $(F_i)_{i\in I}$ (directed here meaning that for any $i$ and $j$, there exists a $k$ such that $F_i\cup F_j\subseteq F_k$), if $K = \bigcup_{i\in I} F_i$, then $K$ is the colimit in the category of fields of the diagram consisting of the $(F_i)_{i\in I}$ and the inclusion maps between them.
In particular, for any field $k$, the algebraic closure $\overline{k}$ is the directed colimit (in the category of fields) of the diagram consisting of all of its subfields which are finite extensions of $k$.
Typically, for categories of algebraic structures, the forgetful functor to $\mathsf{Set}$ preserves directed colimits. So if we have a directed diagram where the arrows are inclusions, the underlying set of the directed colimit will just be the union of the underlying sets of the structures in the diagram.
To make this more precise: For any $\forall\exists$-axiomatizable first-order theory $T$, the category $\text{Mod}(T)$ of models of $T$ and embeddings has directed colimits, and the forgetful functor to $\mathsf{Set}$ preserves directed colimits. And for any positive $\forall\exists$-axiomatizable theory $T$, the category $\text{Mod}(T)$ of models of $T$ and homomorphisms has directed colimits, and the forgetful functor to $\mathsf{Set}$ preserves directed colimits. Note that the category of fields is $\forall\exists$-axiomatizable, and every homomorphism of fields is an embedding.
