Turing degrees of isomorphism types of algebraic objects
Wesley Calvert, Valentina Harizanov, Alexandra Shlapentokh Β· Journal of the London Mathematical Society Β· 2007
The Turing degree spectrum of a countable structure π is the set of all Turing degrees of isomorphic copies of π. The Turing degree of the isomorphism type of π, is the least Turing degree in its degree spectrum. We show that there are structures with isomorphism types of arbitrary Turing degrees in each of the following classes: countable fields, rings, and torsion-free Abelian groups of any finite rank. We also show that there are structures in each of these classes the isomorphism types of which do not have Turing degrees. The case of torsion-free Abelian groups of finite rank settles a question left open by Knight, Downey and Jockusch [Downey, Complexity, logic, and recursion theory, Lecture Notes in Pure and Applied Mathematics 187 (ed. A. Sorbi; Marcel Dekker, New York, 1997) 157β205].