[FOM] Existence of algebraic closures of fields

Wesley Calvert wesley.calvert at murraystate.edu
Tue Mar 2 17:44:20 EST 2010

> 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

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