Can one construct an algebraic closure of fields like $\mathbb{F}_p(T)$ without Zorn's lemma?

Question

I have heard that an algebraic closure of $\mathbb{Q}$ can be constructed without Zorn's lemma and so can an algebraic closure of a finite field $\mathbb{F}_p$. What about $\mathbb{F}_p(T)$? Do there exist fields for which it is not possible to conclude that that there is an algebraic closure without Zorn's lemma?

Answer 1

On a technical note, the existence (and uniqueness) of algebraic closures follow from a weaker assumption than Zorn's lemma.

But we can treat this question as asking "without any appeal to the axiom of choice". Countable fields, and in fact any well-ordered field has a canonical algebraic closure which we can construct.

It may be, however, that there are two non-isomorphic algebraic closures.

Related:

Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?