The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma.  A quick Google search leads to a <a href="http://www3.interscience.wiley.com/journal/113463992/articletext?DOI=10.1002%2Fmalq.19920380136">1992 paper of Banaschewski</a>, which I don't have access to, asserting that the proof only requires the ultrafilter lemma.  Questions:

- Is it known whether the two are equivalent in ZF?
- Would anyone like to give a quick sketch of the construction assuming the ultrafilter lemma?  I dislike the usual construction and am looking for others.