[FOM] Existence of algebraic closures of fields

[Wesley Calvert]

> Rabin's construction of algebraic closures applies only to computable
> fields: what he shows is that the algebraic closure of a cpmputable field F
> can be constructed as a computable extension of F. No field in a model of ZF
> which lacks an algebraic closure can be computable.

Well, it relativizes in a fairly trivial way to show somewhat more than 
that.  It shows that any particular countable field F has an algebraic 
closure computable in F.  I think the barrier may be countability; one 
needs, as best I recall (or at least seems to need), an effective 
indexing of the irreducible polynomials by the field elements (there may 
even be some well-ordering needed), and perhaps that is harder to find 
for, e.g. some size continuum fields.

The Pincus example is interesting to me.  I'll have to look that up.


-- 
Wesley Calvert
Department of Mathematics & Statistics
Faculty Hall 6C
Murray State University
Murray, Kentucky 42071
(270) 809-2503