Real closed exponential fields

In an extended abstract [20], 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. The construction becomes canonical once we fix the real closed exponential field R, a residue field section, and a well ordering <. The construction is clearly constructible over these objects. Each step looks effective, but it may require many steps. We produce an example of an exponential field R with a residue field k and a well ordering < such that D(R) is low and k and < are ∆3, and Ressayre’s construction cannot be completed in LωCK 1 .