On Computable Field Embeddings and Difference Closed Fields
Canadian journal of mathematics, Tome 69 (2017) no. 6, pp. 1338-1363

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2016-044-7
Mots-clés : 03D45, 03C57, 12Y05, computable algebra, algebraic field, difference field, extension of automorphism
Harrison-Trainor, Matthew; Melnikov, Alexander; Miller, Russell. On Computable Field Embeddings and Difference Closed Fields. Canadian journal of mathematics, Tome 69 (2017) no. 6, pp. 1338-1363. doi: 10.4153/CJM-2016-044-7
@article{10_4153_CJM_2016_044_7,
     author = {Harrison-Trainor, Matthew and Melnikov, Alexander and Miller, Russell},
     title = {On {Computable} {Field} {Embeddings} and {Difference} {Closed} {Fields}},
     journal = {Canadian journal of mathematics},
     pages = {1338--1363},
     year = {2017},
     volume = {69},
     number = {6},
     doi = {10.4153/CJM-2016-044-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2016-044-7/}
}
TY  - JOUR
AU  - Harrison-Trainor, Matthew
AU  - Melnikov, Alexander
AU  - Miller, Russell
TI  - On Computable Field Embeddings and Difference Closed Fields
JO  - Canadian journal of mathematics
PY  - 2017
SP  - 1338
EP  - 1363
VL  - 69
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2016-044-7/
DO  - 10.4153/CJM-2016-044-7
ID  - 10_4153_CJM_2016_044_7
ER  - 
%0 Journal Article
%A Harrison-Trainor, Matthew
%A Melnikov, Alexander
%A Miller, Russell
%T On Computable Field Embeddings and Difference Closed Fields
%J Canadian journal of mathematics
%D 2017
%P 1338-1363
%V 69
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2016-044-7/
%R 10.4153/CJM-2016-044-7
%F 10_4153_CJM_2016_044_7

[Bab62] [Bab62] Babbitt, Albert E., Jr., Finitely generated pathological extensions of difference fields. Trans. Amer. Math. Soc. 102(1962), 63–81. Google Scholar | DOI

[Bes40] [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] [CH99] Chatzidakis, Zoé and Hrushovski, Ehud, Model theory of difference fields. Trans. Amer. Math. Soc. 351(1999), no. 8, 2997–3071. Google Scholar | DOI

[Cle70] [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] [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] [Coh65] Cohn, Richard M., Difference algebra. Interscience Publishers John Wiley & Sons, New York, 1965. Google Scholar

[DHS13] [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] [Eva73] Evanovich, Peter, Algebraic extensions of difference fields. Trans. Amer. Math. Soc. 179(1973), 1–22. Google Scholar | DOI

[FJ08] [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] [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] [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] [Har74] Harrington, Leo, Recursively presentable prime models. J. Symbolic Logic 39(1974), 305–309. Google Scholar | DOI

[Har98] [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] [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] [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] [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] [Mal61] Mal'cev, Anatoly I.. Constructive algebras. I. Uspehi Mat. Nauk 16(1961), no. 3 (99),3–60. Google Scholar

[Mil83] [Mil83] Millar, Terrence, Omitting types, type spectrums, and decidability. J. Symbolic Logic 48(1983), no. 1,171–181. Google Scholar | DOI

[MilO8] [MilO8] Miller, Russell, Computable fields and Galois theory. Notices Amer. Math. Soc. 55(2008), no. 7, 798–807. Google Scholar

[Mor53] [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] [Rab60] Rabin, Michael O., Computable algebra, general theory and theory of computable fields. Trans. Amer. Math. Soc. 95(1960), 341–360. Google Scholar

[vdW70] [vdW70] van der Waerden, Bartel L., Algebra. Vol 1. Frederick Ungar, New York ,1970. Google Scholar

Cité par Sources :