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d-computable categoricity for algebraic fields

Published online by Cambridge University Press:  12 March 2014

Russell Miller
Russell Miller*
Affiliation:
Department of Mathematics, Queens College – C.U.N.Y., 65-30 Kissena Blvd., Flushing, New York 11367, USAPh.D. Programs in Computer Science & Mathematics, The Graduate Center of C.U.N.Y., 365 Fifth Avenue, New York, New York 10016, USA, E-mail: Russell.Miller@qc.cuny.edu
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Abstract

We use the Low Basis Theorem of Jockusch and Soare to show that all computable algebraic fields are d-computably categorical for a particular Turing degree d with d′ = 0″, but that not all such fields are 0′-computably categorical. We also prove related results about algebraic fields with splitting algorithms, and fields of finite transcendence degree over ℚ.

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009, pp. 1325 - 1351
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Ash, C.J., Knight, J.F., Manasse, M.S., and Slaman, T.A., Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195–205.CrossRefGoogle Scholar
[2]Calvert, W., Harizanov, V., and Shlapentokh, A., Turing degrees of the isomorphism types of algebraic objects, Journal of London Mathematical Society, vol. 73 (2007), pp. 273–286.CrossRefGoogle Scholar
[3]Cenzer, D., Π10-classes in recursion theory, Handbook of computability theory (Griffor, E.R., editor). Elsevier, Amsterdam, 1999, pp. 37–85.CrossRefGoogle Scholar
[4]Downey, R.G. and Jockusch, C.G. Jr., Every low Boolean algebra is isomorphic to a recursive one. Proceedings of the American Mathematical Society, vol. 122, 1994, pp. 871–880.CrossRefGoogle Scholar
[5]Downey, R.G., Hirschfeldt, D.R., and Khoussainov, B., Uniformity in computable structure theory, Algebra and Logic, vol. 42 (2003), pp. 318–332.CrossRefGoogle Scholar
[6]Edwards, H.M., Galois theory, Springer-Verlag, New York, 1984.Google Scholar
[7]Ershov, Y.L. and Goncharov, S.S., Constructive fields, Constructive models, Kluwer Academic/Plenum Press, New York, 2000, Section 2.5.CrossRefGoogle Scholar
[8]Ershov, Yu.L., Theorie der Numerierungen, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 289–371.Google Scholar
[9]Fokina, E., Kalimullin, I., and Miller, R.G., Degrees of categoricity of computable structures, to appear.Google Scholar
[10]Fried, M.D. and Jarden, M., Field arithmetic, Springer-Verlag, Berlin, 1986.CrossRefGoogle Scholar
[11]Frohlich, A. and Shepherdson, J.C., Effective procedures in field theory. The Philosophical Transactions of the Royal Society, Series A, vol. 248 (1956), no. 950, pp. 407–432.Google Scholar
[12]Goncharov, S.S., Autostability and computable families of constructivizations, Algebra and Logic, vol. 14 (1975), pp. 647–680, (Russian), 392–409 (English translation).CrossRefGoogle Scholar
[13]Goncharov, S.S., Nonequivalent constructivizations, Proc. Math. Inst. Sib. Branch Acad. Sci., Nauka, Novosibirsk, 1982.Google Scholar
[14]Goncharov, S.S., Autostable models and algorithmic dimensions, Handbook of recursive mathematics, vol. 1, Elsevier, Amsterdam, 1998, pp. 261–287.Google Scholar
[15]Goncharov, S.S. and Dzgoev, V.D., Autostability of models, Algebra and Logic, vol. 19 (1980), pp. 45–58, (Russian), 28–37 (English translation).Google Scholar
[16]Goncharov, S.S., Lempp, S., and Solomon, R.. The computable dimension of ordered abelian groups, Advances in Mathematics, vol. 175 (2003), no. 1, pp. 102–143.CrossRefGoogle Scholar
[17]Hirschfeldt, D.R., Khoussainov, B., Shore, R.A., and Slinko, A.M., Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.CrossRefGoogle Scholar
[18]Jacobson, N., Basic algebra I, W.H. Freeman & Co., New York, 1985.Google Scholar
[19]Jockusch, C.G. and Soare, R.I., Π10-classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[20]Jockusch, C.G., Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.CrossRefGoogle Scholar
[21]Khoussainov, B. and Shore, R.A., Effective model theory: The number of models and their complexity, Models and computability: Invited papers from Logic Colloquium '97 (Cooper, S.B. and Truss, J.K., editors), LMSLNS, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193–240.CrossRefGoogle Scholar
[22]Kogabaev, N., Kudinov, O., and Miller, R.G., The computable dimension of I-trees of infinite height, Algebra and Logic, vol. 43 (2004), no. 6, pp. 393–407.CrossRefGoogle Scholar
[23]Kronecker, L., Grundzüge einer arithmetischen Theorie der algebraischen Größen, Journal für die reine undangewandte Mathematik, vol. 92 (1882), pp. 1–122.Google Scholar
[24]Lempp, S., Mccoy, C., Miller, R.G., and Solomon, R., Computable categoricity of trees of finite height, this Journal, vol. 70 (2005), pp. 151–215.Google Scholar
[25]Miller, R.G., The δ20-spectrum of a linear order, this Journal, vol. 66 (2001), pp. 470–486.Google Scholar
[26]Miller, R.G., The computable dimension of trees of infinite height, this Journal, vol. 70 (2005), pp. 111–141.Google Scholar
[27]Miller, R.G. and Schoutens, H., Computably categorical fields via Fermat's Last Theorem, to appear.Google Scholar
[28]Metakides, G.Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289–320.CrossRefGoogle Scholar
[29]Rabin, M., Computable algebra, general theory, and theory of computable fields, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
[30]Remmel, J.B., Recursive isomorphism types of recursive Boolean algebras, this Journal, vol. 46 (1981), pp. 572–594.Google Scholar
[31]Remmel, J.B., Recursively categorical linear orderings, Proceedings of the American Mathematical Society, vol. 83 (1981), pp. 387–391.CrossRefGoogle Scholar
[32]Richter, L.J., Degrees of structures, this Journal, vol. 46 (1981), pp. 723–731.Google Scholar
[33]Soare, R.I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
[34]Stoltenberg-Hansen, V. and Tucker, J.V., Computable rings and fields, Handbook of computability theory (Griffor, E.R., editor), Elsevier, Amsterdam, 1999, pp. 363–447.CrossRefGoogle Scholar
[35]Ventsov, Y.G., Effective choice for relations and reducibilities in classes of constructive and positive models, Algebra and Logic, vol. 31 (1992), pp. 63–73.CrossRefGoogle Scholar
[36]van der Waerden, B.L., Algebra, vol. I, Springer-Verlag, New York, 1970, trans. Blum, F. and Schulenberger, J.R..Google Scholar