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Russell Miller, Alexandra Shlapentokh (2011)
Computable categoricity for algebraic fields with splitting algorithmsTransactions of the American Mathematical Society, 367
R. Soare (1987)
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Reducibility, Degree Spectra, and Lowness in Algebraic Structures
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Computable fields and Galois theory
W. Hodges (1997)
A Shorter Model Theory
E. Fokina, I. Kalimullin, Russell Miller (2010)
Degrees of categoricity of computable structuresArchive for Mathematical Logic, 49
A. Frolov, I. Kalimullin, Russell Miller (2009)
Spectra of Algebraic Fields and Subfields
V. Harizanov, Russell Miller (2007)
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Computability Theory and Applications: The Art of Classical Computability. two volumes
Results of R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. Here, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finite-valence, pointed graphs, and we show that the degree spectrum results do not hold for these trees and graphs. We also offer an explanation for why the degree spectrum results distinguish these classes of structures: although all three structures are algebraic structures, the fields are what we call effectively algebraic.
Archive for Mathematical Logic – Springer Journals
Published: Oct 6, 2012
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