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 \(\overline{F}\) (either \(\overline{\mathbb{Q}}\) or \(\overline{{\mathbb{Z}}/(p)}\)). We consider the Turing degree spectrum of F in both cases, as a structure and as a relation on \(\overline{F}\), 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 \(\overline{F}\).
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Frolov, A., Kalimullin, I., Miller, R. (2009). Spectra of Algebraic Fields and Subfields. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_24
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DOI: https://doi.org/10.1007/978-3-642-03073-4_24
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