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Preprint Number 401
401. Paola D'Aquino, Julia F. Knight, Salma Kuhlmann, and Karen Lange Real closed exponential fields E-mail: Submission date: 17 December 2011. Abstract: 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 D^c(R) is low and k and < are Δ^0_3, and Ressayre's construction cannot be completed in L_{ω_1^{CK}}. Mathematics Subject Classification: Primary 03C64, Secondary 03D45, 03C57, 12J10, 12J15, 14P99 Keywords and phrases: |

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