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ELEMENTARY EQUIVALENCE THEOREM FOR PAC STRUCTURES

Part of:Model theory

Published online by Cambridge University Press:  26 October 2020

JAN DOBROWOLSKI,
DANIEL MAX HOFFMANN and
JUNGUK LEE
JAN DOBROWOLSKI
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI WROCŁAW, POLANDSCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS LEEDS, UK E-mail: dobrowol@math.uni.wroc.pl E-mail: J.Dobrowolski@leeds.ac.uk URL: http://www.math.uni.wroc.pl/~dobrowol/
DANIEL MAX HOFFMANN
Affiliation:
INSTYTUT MATEMATYKI UNIWERSYTET WARSZAWSKI WARSZAWA, POLANDDEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN USA E-mail: daniel.max.hoffmann@gmail.com URL: https://sites.google.com/site/danielmaxhoffmann/
JUNGUK LEE
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI WROCŁAW, POLANDDEPARTMENT OF MATHEMATICAL SCIENCES KAIST, 291, DAEHAK-RO, YUSEONG-GU, DAEJEON, 34141 REPUBLIC OF KOREA E-mail: ljwhayo@kaist.ac.kr URL: https://sites.google.com/site/leejunguk0323/
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Abstract

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We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism type of their absolute Galois groups. Our results concern two cases: saturated PAC structures and nonsaturated PAC structures.

Keywords

pseudo-algebraically closed structuresGalois groups

MSC classification

Primary:03C95: Abstract model theory
Secondary:03C45: Classification theory, stability and related concepts03C50: Models with special properties (saturated, rigid, etc.)

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020, pp. 1467 - 1498
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Copyright
© The Association for Symbolic Logic 2020

References

REFERENCES

Ax, J., Solving diophantine problems modulo every prime. Annals of Mathematics, vol. 85 (1967), no. 2, pp. 161–183.CrossRefGoogle Scholar
Ax, J., The elementary theory of finite fields. Annals of Mathematics, vol. 88 (1968), no. 2, pp. 239–271.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Model Theory. North-Holland, Amsterdam, 1990.Google Scholar
Chatzidakis, Z., Properties of forking in ω-free pseudo-algebraically closed fields, this Journal, vol. 67 (2002), no. 3, pp. 957–996.Google Scholar
Chatzidakis, Z., Amalgamation of types in pseudo-algebraically closed fields and applications. Journal of Mathematical Logic, vol. 19 (2019), no. 2, 1950006.CrossRefGoogle Scholar
Chatzidakis, Z. and Hrushovski, E., Perfect pseudo-algebraically closed fields are algebraically bounded. Journal of Algebra, vol. 271 (2004), no. 2, pp. 627–637.CrossRefGoogle Scholar
Chatzidakis, Z. and Pillay, A., Generic structures and simple theories. Annals of Pure and Applied Logic, vol. 95 (1998), no. 1-3, pp. 71–92.CrossRefGoogle Scholar
Cherlin, G., van den Dries, L., and Macintyre, A., The elementary theory of regularly closed fields, preprint, 1980. Available at http://sites.math.rutgers.edu/~cherlin/Preprint/CDM2.pdf.Google Scholar
Cherlin, G., van den Dries, L., and Macintyre, A., Decidability and undecidability theorems for PAC-fields. Bulletin of the American Mathematical Society, vol. 4 (1981), no. 1, pp. 101–104.Google Scholar
Ershov, Y., Regularly closed fields. Soviet Mathematics: Doklady, vol. 21 (1980), pp. 510–512.Google Scholar
Frey, G., Pseudo algebraically closed fields with non-archimedean real valuations. Journal of Algebra, vol. 26 (1973), no. 2, pp. 202–207.CrossRefGoogle Scholar
Fried, M. D. and Jarden, M., Field Arithmetic, A Series of Modern Surveys in Mathematics, Springer, New York, 2008.Google Scholar
Hoffmann, D. M., Model theoretic dynamics in Galois fashion. Annals of Pure and Applied Logic, 170 (2019), no. 7, pp. 755–804.CrossRefGoogle Scholar
Hoffmann, D. M., On galois groups and PAC substructures. Fundamenta Mathematicae, vol. 250 (2020), pp. 151–177.CrossRefGoogle Scholar
Hoffmann, D. M. and Lee, J., Co-theory of sorted profinite groups for PAC structures. Available at https://arxiv.org/abs/1905.09748.Google Scholar
Hrushovski, E., Pseudo-finite fields and related structures, Model Theory and Applications (Bȳlair, L. et al., editors), Quaderni di Matematics, vol. 11, Aracne, Rome, 2005, p. 151–212.Google Scholar
Jarden, M. and Kiehne, U., The elementary theory of algebraic fields of finite corank. Inventiones Mathematicae, vol. 30 (1975), no. 3, pp. 275–294.CrossRefGoogle Scholar
Johnson, W., On the proof of elimination of imaginaries in algebraically closed valued fields. Available at https://arxiv.org/pdf/1406.3654.pdf.Google Scholar
Kaplan, I. and Ramsey, N., On Kim-independence. Journal of the European Mathematical Society, vol. 22 (2020), pp. 1423–1474.CrossRefGoogle Scholar
Keisler, H. J., The ultraproduct construction. Available at http://logic.amu.edu.pl/images/1/14/Keislerultraproduct.pdf.Google Scholar
Pillay, A. and Polkowska, D., On PAC and bounded substructures of a stable structure, this Journal, vol. 71(2006), no. 2, pp. 460–472.Google Scholar
Polkowska, O.P. N. M., On simplicity of bounded pseudoalgebraically closed structures. Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 173–193.CrossRefGoogle Scholar
Ribes, L. and Zalesskii, P., Profinite Groups. Springer, New York, 2000.CrossRefGoogle Scholar