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

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?

Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?
• Jessica, the answer is in the context where the axiom of choice fails. It might be the case where $\Bbb Q$ has two non-isomorphic algebraic closures. If we have some choice then we can prove this never happens, but if we don't have choice at all it might be that this is what happens in the universe. – Asaf Karagila Mar 12 '13 at 16:43