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Categoricity properties for computable algebraic fields


Authors: Denis R. Hirschfeldt, Ken Kramer, Russell Miller and Alexandra Shlapentokh
Journal: Trans. Amer. Math. Soc. 367 (2015), 3981-4017
MSC (2010): Primary 03D45; Secondary 03C57, 12L99
Published electronically: October 20, 2014
MathSciNet review: 3324917
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Abstract: We examine categoricity issues for computable algebraic fields. We give a structural criterion for relative computable categoricity of these fields, and use it to construct a field that is computably categorical, but not relatively computably categorical. Finally, we show that computable categoricity for this class of fields is $ \Pi ^0_4$-complete.


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Additional Information

Denis R. Hirschfeldt
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: drh@math.uchicago.edu

Ken Kramer
Affiliation: Department of Mathematics, Queens College – C.U.N.Y., 65-30 Kissena Boulevard, Flushing, New York 11367 — and — Ph.D. Program in Mathematics, C.U.N.Y. Graduate Center, 365 Fifth Avenue, New York, New York 10016
Email: kkramer@qc.cuny.edu

Russell Miller
Affiliation: Department of Mathematics, Queens College – C.U.N.Y., 65-30 Kissena Boulevard, Flushing, New York 11367 — and — Ph.D. Programs in Mathematics and Computer Science, C.U.N.Y. Graduate Center, 365 Fifth Avenue, New York, New York 10016
Email: Russell.Miller@qc.cuny.edu

Alexandra Shlapentokh
Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email: shlapentokha@ecu.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06094-7
Received by editor(s): November 3, 2011
Received by editor(s) in revised form: January 29, 2013
Published electronically: October 20, 2014
Additional Notes: The first author was partially supported by Grants # DMS–0801033 and DMS–1101458 from the National Science Foundation
The second author was partially supported by Grant # DMS–0739346 from the National Science Foundation and by Grant #44-459 from the PSC-CUNY Research Award Program
The third author was partially supported by Grant # DMS–1001306 from the National Science Foundation, by Grant # 13397 from the Templeton Foundation, by the Centre de Recerca Matemática, and by several grants from The City University of New York PSC-CUNY Research Award Program
The fourth author was partially supported by Grants # DMS–0650927 and DMS–1161456 from the National Science Foundation, by Grant # 13419 from the Templeton Foundation, and by an ECU Faculty Senate Summer 2011 Grant
Article copyright: © Copyright 2014 American Mathematical Society