Degree spectra of real closed fields

Several researchers have recently established that for every Turing degree $$\varvec{c}$$c, the real closed field of all $$\varvec{c}$$c-computable real numbers has spectrum $$ \{ \varvec{d} : \varvec{d}'\ge \varvec{c}'' \} $${d:d′≥c′′}. We investigate the spectra of real closed fields further, focusing first on subfields of the field $$\mathbb {R}_{\varvec{0}}$$R0 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 $$\mathbb {R}_{\varvec{0}}$$R0 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.