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On Computable Field Embeddings and Difference Closed Fields

Published online by Cambridge University Press:  20 November 2018

Matthew Harrison-Trainor,
Alexander Melnikov and
Russell Miller
Matthew Harrison-Trainor
Affiliation:
Group in Logic and the Methodology of Science, University of California, Berkeley, USA e-mail: matthew.h-t@berkeley.edu
Alexander Melnikov
Affiliation:
The Institute of Natural and Mathematical Science, Massey University, New Zealand e-mail: alexander.g.melnikov@gmail.com
Russell Miller
Affiliation:
Dept. of Mathematics, Queens College, - Ph.D. Programs inMathematics - Computer Science, Graduate Center, City University of New York, USA e-mail: Russell.Miller@qc.cuny.edu
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Abstract

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We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of computable difference fields into computable difference closed fields.

Keywords

03D4503C5712Y05computable algebraalgebraic fielddifference fieldextension of automorphism

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 6 , 01 December 2017, pp. 1338 - 1363
Copyright
Copyright © Canadian Mathematical Society 2016

References

[Bab62] Babbitt, Albert E., Jr., Finitely generated pathological extensions of difference fields. Trans. Amer. Math. Soc. 102(1962), 63–81.http://dx.doi.org/10.1090/S0002-9947-1962-0133326-0 Google Scholar
[Bes40] Besicovitch, Abram S.,On the linear independence of fractional powers of integers. J. London Math. Soc. 15(1940), 3–6,. http://dx.doi.Org/10.1112/jlms/sl -15.1.3 Google Scholar
[CH99] Chatzidakis, Zoé and Hrushovski, Ehud, Model theory of difference fields. Trans. Amer. Math. Soc. 351(1999), no. 8, 2997–3071.http://dx.doi.org/10.1090/S0002-9947-99-02498-8 Google Scholar
[Cle70] Cleave, John P. , Some properties of recursively inseparable sets. Z. Math. Logik Grundlagen Math. 16(1970), 187–200.http://dx.doi.Org/10.1002/malq.1 9700160208 Google Scholar
[Coh52] Cohn, Richard M., Extensions of difference fields. Amer. J. Math. 74(1952) 507–530. http://dx.doi.Org/10.2307/2372012 Google Scholar
[Coh65] Cohn, Richard M., Difference algebra. Interscience Publishers John Wiley & Sons, New York, 1965.Google Scholar
[DHS13] Dorais, Franois G., Jeffry Hirst, and Paul Shafer, Reverse mathematics and algebraic field extensions. Computability 2(2013), no. 2, 75–92.Google Scholar
[Eva73] Evanovich, Peter, Algebraic extensions of difference fields. Trans. Amer. Math. Soc. 179(1973), 1–22.http://dx.doi.org/10.1090/S0002-9947-1973-0314809-4 Google Scholar
[FJ08] Fried, Michael D.and Jarden, Moshe, Field arithmetic. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, 11. Springer-Verlag, Berlin, 2008.Google Scholar
[FSS83] Friedman, Harvey M., Simpson, Stephen G., and Smith, Rick L., Countable algebra and set existence axioms. Ann. Pure Appl. Logic 25(1983), no. 2,141–181.http://dx.doi.Org/10.1016/0168-0072(83)90012-X Google Scholar
[Gou89] Goursat, Edouard, Sur les substitutions orthogonales et les divisions réguli ères de l'espace. Ann. Sci. Ècole Norm. Sup. (3) 6(1889), 9–102.Google Scholar
[Har74] Harrington, Leo, Recursively presentable prime models. J. Symbolic Logic 39(1974), 305–309.http://dx.doi.org/10.2307/2272643 Google Scholar
[Har98] Harizanov, Valentina S., Pure computable model theory. In: Handbook of recursive mathematics, Vol. 1. Stud. Logic Found. Math., 138. North-Holland, Amsterdam, 1998, pp. 3–114.Google Scholar
[HTMM15] Harrison-Trainor, Matthew, Alexander Melnikov, and Antonio Montalbân, Independence in computable algebra. J. Algebra 443(2015), 441–468. http://dx.doi.Org/10.101 6/j.jalgebra.2O1 5.06.004 Google Scholar
[Kro82] Kronecker, Leopold, Grundzuge einer arithmetischen Théorie der algebraischen Grofien. J. Reine Angew. Math. 92(1882), 1–122.http://dx.doi.Org/10.1515/crll.1882.92.1 Google Scholar
[Mac97] Macintyre, Angus, Generic automorphisms of fields. Ann. Pure Appl. Logic 88(1997), no. 2-3,165–180. http://dx.doi.Org/1 0.1016/S0168-0072(97)00020-1 Google Scholar
[Mal61] Mal'cev, Anatoly I.. Constructive algebras. I. Uspehi Mat. Nauk 16(1961), no. 3 (99),3–60.Google Scholar
[Mil83] Millar, Terrence, Omitting types, type spectrums, and decidability. J. Symbolic Logic 48(1983), no. 1,171–181.http://dx.doi.org/10.2307/2273331 Google Scholar
[MilO8] Miller, Russell, Computable fields and Galois theory. Notices Amer. Math. Soc. 55(2008), no. 7, 798–807.Google Scholar
[Mor53] Mordell, Louis J., On the linear independence of algebraic numbers. Pacific J. Math. 3(1953), 625–630.http://dx.doi.Org/10.2140/pjm.1953.3.625 Google Scholar
[Rab60] Rabin, Michael O., Computable algebra, general theory and theory of computable fields. Trans. Amer. Math. Soc. 95(1960), 341–360.Google Scholar
[vdW70] van der Waerden, Bartel L., Algebra. Vol 1. Frederick Ungar, New York ,1970.Google Scholar