Re: Construction of a real closure

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