[FOM] Existence of algebraic closures of fields

Wesley Calvert wesley.calvert at murraystate.edu
Mon Mar 1 16:53:45 EST 2010

> 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
Faculty Hall 6C
Murray State University
Murray, Kentucky 42071
(270) 809-2503

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