[FOM] Algebraic closure of Q

[Timothy Y. Chow]

Stephen G Simpson <simpson at math.psu.edu> wrote:
> Therefore, it appears that Hodges must be talking about some other
> context and/or some other sense of algebraic closure.  I am not
> familiar with Hodges' result, so could you please explain the context.
> 
> Define an algebraic closure of a countable field K to be a countable
> algebraically closed field L together with a monomorphism h: K --> L
> such that every element of L is a root of a nonzero polynomial whose
> coefficients are images under h of elements of K.

I believe that the difference between your definition and the definition 
used by Hodges is that you require the algebraic closure of a countable 
field to be countable *by definition*, whereas Hodges takes an algebraic 
closure of K to be any extension of K that is (1) algebraically closed and 
(2) contains no proper subfield that is algebraically closed.  If a 
countable union of finite sets is not necessarily countable, then an 
algebraic closure (in the sense of Hodges) of a countable field K need not 
be countable.

> Then, the existence of an algebraic closure of any countable field K is 
> provable in RCA_0. The uniqueness, up to isomorphism over K, is 
> equivalent over RCA_0 to WKL_0.
> 
> Define a strong algebraic closure of a countable field K to be an
> algebraic closure h: K --> L as above, such that the image of K under
> h exists as a subfield of L.  Then, the existence of a strong
> algebraic closure of any countable field is equivalent over RCA_0 to
> ACA_0.  The uniqueness is provable in WKL_0, hence in ACA_0.

Thanks for the information!

I wrote:
> So I think I see how it suffices to assume that a countable union of 
> *countable* sets is countable.  How do you get this down to a countable 
> union of *finite* sets?

Joe Shipman explained this to me.  One really only needs to adjoin the 
roots of each polynomial at each step, not the whole field generated by 
them.  One still gets a field anyway, as one can prove separately.

Tim