[FOM] Existence of algebraic closures of fields

jbell@uwo.ca jbell at uwo.ca
Tue Mar 2 13:40:12 EST 2010

In his 1972 JSL paper "Zermelo-Fraenkel consistencies by
Fraenkel-Mostowski methods", David Pincus showed inter alia that there is a 
model of ZF in which not every field has an algebraic closure. (The ZF-model 
is constructed from the basic Fraenkel ZFA- model.) This result also appears 
as Theorem 10.13 in Jech's 1973 book "The Axiom of Choice."

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.

- John Bell

Professor John L. Bell
Department of Philosophy
University of Western Ontario
London, Ontario N6A 3K7


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