Spectra of Algebraic Fields and Subfields

Authors:
Andrey Frolov;Iskander Kalimullin;Russell Miller
Affiliations:
N.G. Chebotarev Research Institute of Mechanics and Mathematics, Kazan State University, Kazan, Russia 420008;N.G. Chebotarev Research Institute of Mechanics and Mathematics, Kazan State University, Kazan, Russia 420008;Queens College of CUNY, Flushing, USA 11367 and The CUNY Graduate Center, New York, USA 10016
Venue:
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Year:
2009

Citing 2
Cited 2

Recursively enumerable sets and degrees

Recursively enumerable sets and degrees
Spectra of Algebraic Fields and Subfields

CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice

Spectra of Algebraic Fields and Subfields

CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Effective algebraicity

Archive for Mathematical Logic

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Abstract

An algebraic field extension of *** or ***/(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 $\overline{F}$.