RE: Construction of a real closure

[Michael Barr <mbarr@math.mcgill.ca>]

I don't think this is what I was asking, although it may be related.
Certainly, the context I have is one in which if you have a real closure
and adjoin i, you get an algebraic closure.  And the real closure is
unique because it is the usual real closure.  In fact, the usual real
closure is used to prove some things.  As I said, the construction is
constructive; the proofs are classical.

Let me give an example of the flavor.  Suppose you want to have a square
root of a > 0.  It is decidable (by hypothesis) if a > 0; what may not be
decidable is whether a has a square root.  What the student did (he
credits Joyal with some of the main ideas, BTW, and this may be one of
them) was to form F[x]/I where I is the ideal of all polynomials that
vanish at sqrt(a).  Even though it may not be decidable if sqrt(a) in F,
it is still decidable if a polynomial vanishes there.  First use Sturm's
criterion to find an interval that contains exactly one root of f(x) = x^2
- a (in a real closure) and then for any polynomial g, g is in I iff
gcd(f,g) has a root in that interval, again using Sturm's criterion.
Clearly, F[x]/I contains a square root of a, even if it is not decidable
whether that field is F.

Michael

On Thu, 4 May 2006, Marta Bunge wrote:

>
> Dear Michael,
>
> >Is there a reference for the fact that a countable decidably ordered field
> >has a constructable (and decidably ordered) real closure?
> >
>
> In my paper "Sheaves and Prime Model Extensions", J. of Algebra 68 (1981)
> 79-96, there is a proof of the existence of the real closure of an ordered
> field in any elementary topos, plus considerations about the failure of the
> existence of the algebraic closure. The context is more general (model
> theory in toposes), and there are other instances which I do not recall
> offhand. Maybe that is not what you are asking? I thought that I would
> mention it, just in case.
>
> Best,
> Marta
>
>