Archive for Mathematical Logic 58 (3-4):387-411 (2019)

Several researchers have recently established that for every Turing degree \, the real closed field of all \-computable real numbers has spectrum \. We investigate the spectra of real closed fields further, focusing first on subfields of the field \ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of \ with no computable copy. Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation.
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DOI 10.1007/s00153-018-0638-z
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Degrees Coded in Jumps of Orderings.Julia F. Knight - 1986 - Journal of Symbolic Logic 51 (4):1034-1042.

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