Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
A shorter model theory
Computability, definability, categoricity, and automorphisms
Computability, definability, categoricity, and automorphisms
Spectra of Algebraic Fields and Subfields
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
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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.