# Degree spectra of real closed fields

## Russell Miller

### City University of New York

The degree spectrum of a countable structure is the set of all Turing degrees of isomorphic copies of that structure. This topic has been widely studied in computable model theory. Here we examine the possible degree spectra of real closed fields, finding them to offer far more complexity than the related theory of algebraically closed fields. The co-author of this project, Victor Ocasio Gonzalez, showed in his dissertation that, for every linear order *L*, there exists a real closed field whose spectrum is the pre-image under jump of the spectrum of *L*. We add further results, distinguishing the cases of archimedean and non-archimedean real closed fields, and splitting the latter into two subcases based on the existence of a least multiplicative class of positive nonstandard elements. If such a class exists, then finiteness in the field is always decidable, but the case with no such class proves more interesting.

Russell Miller is professor of mathematics at Queens College of CUNY and also at the CUNY Graduate Center. He conducts research in mathematical logic, especially computability theory and its interaction with other areas of mathematics, as in computable model theory. He received his doctorate from the University of Chicago in 2000, as a student of Robert Soare, and subsequently held a postdoctoral position at Cornell University until 2003, when he came to CUNY.