d-computable categoricity for algebraic fields
Russell Miller
Source: J. Symbolic Logic Volume 74, Issue 4 (2009), 1325-1351.
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.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1254748694
Digital Object Identifier: doi:10.2178/jsl/1254748694
References
C.J. Ash, J.F. Knight, M.S. Manasse, and T.A. Slaman, Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195--205.
Mathematical Reviews (MathSciNet):
MR998606
Zentralblatt MATH:
0678.03012
Digital Object Identifier: doi:10.1016/0168-0072(89)90015-8
W. Calvert, V. Harizanov, and A. Shlapentokh, Turing degrees of the isomorphism types of algebraic objects, Journal of London Mathematical Society, vol. 73 (2007), pp. 273--286.
Mathematical Reviews (MathSciNet):
MR2340227
Zentralblatt MATH:
1124.03014
Digital Object Identifier: doi:10.1112/jlms/jdl012
D. Cenzer, $\Pi^0_1$-classes in recursion theory, Handbook of computability theory (E.R. Griffor, editor), Elsevier, Amsterdam, 1999, pp. 37--85.
Mathematical Reviews (MathSciNet):
MR1720779
Zentralblatt MATH:
0939.03047
Digital Object Identifier: doi:10.1016/S0049-237X(99)80018-4
R.G. Downey and Jr. C.G. Jockusch, Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122, 1994, pp. 871--880.
Mathematical Reviews (MathSciNet):
MR1203984
Zentralblatt MATH:
0820.03019
Digital Object Identifier: doi:10.2307/2160766
R.G. Downey, D.R. Hirschfeldt, and B. Khoussainov, Uniformity in computable structure theory, Algebra and Logic, vol. 42 (2003), pp. 318--332.
Mathematical Reviews (MathSciNet):
MR2025716
H.M. Edwards, Galois theory, Springer-Verlag, New York, 1984.
Mathematical Reviews (MathSciNet):
MR743418
Y.L. Ershov and S.S. Goncharov, Constructive fields, Constructive models, Kluwer Academic/Plenum Press, New York, 2000, Section 2.5.
Yu.L. Ershov, Theorie der Numerierungen, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 289--371.
Mathematical Reviews (MathSciNet):
MR439603
E. Fokina, I. Kalimullin, and R.G. Miller, Degrees of categoricity of computable structures, to appear.
M.D. Fried and M. Jarden, Field arithmetic, Springer-Verlag, Berlin, 1986.
Mathematical Reviews (MathSciNet):
MR868860
A. Frohlich and J.C. Shepherdson, Effective procedures in field theory, The Philosophical Transactions of the Royal Society, Series A, vol. 248 (1956), no. 950, pp. 407--432.
Mathematical Reviews (MathSciNet):
MR74349
Digital Object Identifier: doi:10.1098/rsta.1956.0003
JSTOR: links.jstor.org
S.S. Goncharov, Autostability and computable families of constructivizations, Algebra and Logic, vol. 14 (1975), pp. 647--680, (Russian), 392--409 (English translation).
Mathematical Reviews (MathSciNet):
MR437335
--------, Nonequivalent constructivizations, Proc. Math. Inst. Sib. Branch Acad. Sci., Nauka, Novosibirsk, 1982.
Zentralblatt MATH:
0770.03012
--------, Autostable models and algorithmic dimensions, Handbook of recursive mathematics, vol. 1, Elsevier, Amsterdam, 1998, pp. 261--287.
Mathematical Reviews (MathSciNet):
MR1673641
Zentralblatt MATH:
0958.03030
Digital Object Identifier: doi:10.1016/S0049-237X(98)80007-4
S.S. Goncharov and V.D. Dzgoev, Autostability of models, Algebra and Logic, vol. 19 (1980), pp. 45--58, (Russian), 28-37 (English translation).
Mathematical Reviews (MathSciNet):
MR604657
S.S. Goncharov, S. Lempp, and R. Solomon, The computable dimension of ordered abelian groups, Advances in Mathematics, vol. 175 (2003), no. 1, pp. 102--143.
Mathematical Reviews (MathSciNet):
MR1970243
Zentralblatt MATH:
1031.03058
Digital Object Identifier: doi:10.1016/S0001-8708(02)00042-7
D.R. Hirschfeldt, B. Khoussainov, R.A. Shore, and A.M. Slinko, Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71--113.
Mathematical Reviews (MathSciNet):
MR1897023
Zentralblatt MATH:
1016.03034
Digital Object Identifier: doi:10.1016/S0168-0072(01)00087-2
N. Jacobson, Basic algebra I, W.H. Freeman & Co., New York, 1985.
Mathematical Reviews (MathSciNet):
MR780184
C.G. Jockusch and R.I. Soare, $\Pi^0_1$-classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
Mathematical Reviews (MathSciNet):
MR316227
Zentralblatt MATH:
0262.02041
Digital Object Identifier: doi:10.2307/1996261
JSTOR: links.jstor.org
--------, Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39--64.
Mathematical Reviews (MathSciNet):
MR1104053
Zentralblatt MATH:
0734.03026
Digital Object Identifier: doi:10.1016/0168-0072(91)90038-N
B. Khoussainov and R.A. Shore, Effective model theory: The number of models and their complexity, Models and computability: Invited papers from Logic Colloquium '97 (S.B. Cooper and J.K. Truss, editors), LMSLNS, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193--240.
Mathematical Reviews (MathSciNet):
MR1721168
Zentralblatt MATH:
0940.03045
N. Kogabaev, O. Kudinov, and R.G. Miller, The computable dimension of $I$-trees of infinite height, Algebra and Logic, vol. 43 (2004), no. 6, pp. 393--407.
Mathematical Reviews (MathSciNet):
MR2135388
L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraischen Größ en, Journal für die reine und angewandte Mathematik, vol. 92 (1882), pp. 1--122.
S. Lempp, C. McCoy, R.G. Miller, and R. Solomon, Computable categoricity of trees of finite height, Journal of Symbolic Logic, vol. 70 (2005), pp. 151--215.
Mathematical Reviews (MathSciNet):
MR2119128
Zentralblatt MATH:
1104.03026
Digital Object Identifier: doi:10.2178/jsl/1107298515
Project Euclid: euclid.jsl/1107298515
R.G. Miller, The $\Delta^0_2$-spectrum of a linear order, Journal of Symbolic Logic, vol. 66 (2001), pp. 470--486.
Mathematical Reviews (MathSciNet):
MR1833459
Zentralblatt MATH:
0992.03050
Digital Object Identifier: doi:10.2307/2695025
JSTOR: links.jstor.org
--------, The computable dimension of trees of infinite height, Journal of Symbolic Logic, vol. 70 (2005), pp. 111--141.
Mathematical Reviews (MathSciNet):
MR2119126
Zentralblatt MATH:
1098.03049
Digital Object Identifier: doi:10.2178/jsl/1107298513
Project Euclid: euclid.jsl/1107298513
R.G. Miller and H. Schoutens, Computably categorical fields via Fermat's Last Theorem, to appear.
G. Metakides A. Nerode, Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289--320.
Mathematical Reviews (MathSciNet):
MR556895
Zentralblatt MATH:
0469.03028
Digital Object Identifier: doi:10.1016/0003-4843(79)90011-1
M. Rabin, Computable algebra, general theory, and theory of computable fields, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341--360.
Mathematical Reviews (MathSciNet):
MR113807
Zentralblatt MATH:
0156.01201
Digital Object Identifier: doi:10.2307/1993295
J.B. Remmel, Recursive isomorphism types of recursive Boolean algebras, Journal of Symbolic Logic, vol. 46 (1981), pp. 572--594.
Mathematical Reviews (MathSciNet):
MR627907
Zentralblatt MATH:
0543.03031
Digital Object Identifier: doi:10.2307/2273757
JSTOR: links.jstor.org
--------, Recursively categorical linear orderings, Proceedings of the American Mathematical Society, vol. 83 (1981), pp. 387--391.
Mathematical Reviews (MathSciNet):
MR624937
Zentralblatt MATH:
0493.03022
Digital Object Identifier: doi:10.2307/2043534
JSTOR: links.jstor.org
L.J. Richter, Degrees of structures, Journal of Symbolic Logic, vol. 46 (1981), pp. 723--731.
Mathematical Reviews (MathSciNet):
MR641486
Zentralblatt MATH:
0512.03024
Digital Object Identifier: doi:10.2307/2273222
JSTOR: links.jstor.org
R.I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.
Mathematical Reviews (MathSciNet):
MR882921
V. Stoltenberg-Hansen and J.V. Tucker, Computable rings and fields, Handbook of computability theory (E.R. Griffor, editor), Elsevier, Amsterdam, 1999, pp. 363--447.
Mathematical Reviews (MathSciNet):
MR1720739
Zentralblatt MATH:
0944.03040
Digital Object Identifier: doi:10.1016/S0049-237X(99)80028-7
Y.G. Ventsov, Effective choice for relations and reducibilities in classes of constructive and positive models, Algebra and Logic, vol. 31 (1992), pp. 63--73.
Mathematical Reviews (MathSciNet):
MR1289026
B.L. van der Waerden, Algebra, vol. I, Springer-Verlag, New York, 1970, trans. F. Blum and J.R. Schulenberger.
Mathematical Reviews (MathSciNet):
MR1080172
Zentralblatt MATH:
0039.00902
