Real closed exponential fields

Ressayre considered real closed exponential fields and exponential integer parts; i.e., integer parts that respect the exponential function. In [23], 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 and then analyze the complexity. Ressayre's construction is canonical once we fix the real closed exponential field R, a residue field section, and a well ordering < on R. The construction is clearly constructible over these objects. Each step looks effective, but there may be many steps. We produce an example of an exponential field R with a residue field section k and a well ordering < on R such that D^c(R) is low and k and < are \Delta_0^3 , and Ressayre's construction cannot be completed in L_{\omega_1}^{CK}