In an extended abstract Ressayre considered real closed exponential fields
and integer parts that respect the exponential function. He outlined a proof
that every real closed exponential field has an exponential integer part. In
the present paper, we give a detailed account of Ressayre's construction, which
becomes canonical once we fix the real closed exponential field, a residue
field section, and a well ordering of the field. The procedure is constructible
over these objects; each step looks effective, but may require many steps. We
produce an example of an exponential field R with a residue field k and a
well ordering < such that Dc(R) is low and k and < are Ξ30β,
and Ressayre's construction cannot be completed in LΟ1CKββ.Comment: 24 page