[FOM] Existence of algebraic closures of fields

[Wesley Calvert]

> Does the theorem that every field has an algebraic closure require the
> full strength of the axiom of choice?  I mean relative to ZF, is that
> theorem equivalent to AC?  The proofs I know of all refer to Zorn's
> lemma or existence of maximal ideas in ring, or the like.  But did
> they need all of that?

Rabin's effective proof (Trans AMS 95 (1960) pp. 341--360) certainly 
uses quite a lot less than AC.


-- 
Wesley Calvert
Department of Mathematics & Statistics
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Murray State University
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