[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).
-wb
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