[FOM] Existence of algebraic closures of fields
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.
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