[FOM] Existence of algebraic closures of fields

[William Boshuck]

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