[FOM] Existence of algebraic closures of fields

William Boshuck boshuk at math.mcgill.ca
Mon Mar 1 21:37:51 EST 2010

On Sun, Feb 28, 2010 at 09:51:31AM -0500, Colin McLarty wrote:
> 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?

For countable fields one can prove the existence of
an algebraic closure in a weak sense in RCA_0, and
in a stronger sense in ACA_0.  These results can
be found in the second and third chapters of Stephen
Simpson's Subsystems of Second Order Arithmetic (and
generalize an earlier result of Rabin).


More information about the FOM mailing list