Spectra of algebraic fields and subfields

Abstract

An algebraic field extension of ℚ or ℤ/(p) may be regarded either as a structure in its own right, or as a subfield of its algebraic closure F (either ℚ or ℤ/(p)). We consider the Turing degree spectrum of F in both cases, as a structure and as a relation on F, and characterize the sets of Turing degrees that are realized as such spectra. The results show a connection between enumerability in the structure F and computability when F is seen as a subfield of F. © 2009 Springer Berlin Heidelberg.