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SEPARABLY CLOSED VALUED FIELDS: QUANTIFIER ELIMINATION

Published online by Cambridge University Press:  12 August 2016

JIZHAN HONG
JIZHAN HONG*
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES HUAQIAO UNIVERSITY 269 CHENGHUA RD. NORTH QUANZHOU FUJIAN 362021, CHINAE-mail: [email protected]
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Abstract

It is proved in this article that the theory of separably closed nontrivially valued fields of characteristic p > 0 and imperfection degree e > 0 (e ≤ ∞) has quantifier elimination in the language ${{\cal L}_{p,{\rm{div}}}} = \{ + , - , \times ,0,1\} \cup {\{ {\lambda _{n,j}}(x;{y_1}, \ldots ,{y_n})\} _{0 \le n < \omega ,0 \le j < {p^n}}} \cup \{ |\}$; in particular, when e is finite, the corresponding theory has quantifier elimination in the language ${\cal L} = \{ + , - , \times ,0,1\} \cup \{ {b_1}, \ldots ,{b_e}\} \cup {\{ {\lambda _{e,j}}(x;{b_1}, \ldots ,{b_e})\} _{0 \le j < {p^e}}} \cup \{ |\}$.

Keywords

separably closed valued fieldsquantifier eliminationinfinite imperfection degree
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 887 - 900
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Bélair, L., and Point, F., Separably closed fields and contractive Ore modules, preprint on arXiv at http://arxiv.org/abs/1405.1772, 2014.Google Scholar
Delon, F., Quelques propriétés des corps valués en théorie de modèles , Thèse d’état, 1982.Google Scholar
Engler, A. J., and Prestel, A., Valued Fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.Google Scholar
Eršov, J. L., Fields with a solvable theory . Doklady Akademii Nauk SSSR, vol. 174, pp. 19–20, 1967.Google Scholar
Fried, M. D., and Jarden, M., Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, [Results in Mathematics and Related Areas. 3rd Series.], vol. 11, Springer-Verlag, Berlin, 2008. Third edition.Google Scholar
Hong, J., Immediate expansions by valuation of fields, Ph.D. thesis, McMaster University, 2013.Google Scholar
Hrushovski, E., The Mordell-Lang conjecture for function fields . Journal of the American Mathematical Society, vol. 9 (1996), no. 3, pp. 667–690.Google Scholar
Kuhlmann, F.-V., and Pal, K., The model theory of separably tame valued fields, preprint on arXiv http://arxiv.org/abs/1409.3141, 2014.Google Scholar
Mac Lane, S., Modular fields. I. Separating transcendence bases . Duke Mathematical Journal, vol. 5 (1939), no. 2, pp. 372–393.Google Scholar
Macintyre, A., McKenna, K., and van den Dries, L., Elimination of quantifiers in algebraic structures . Advances in Mathematics, vol. 47 (1983), no. 1, pp. 74–87.Google Scholar
Robinson, A., Complete Theories , North-Holland, Amsterdam, second edition. With a preface by H. J. Keisler, Studies in Logic and the Foundations of Mathematics, 1977.Google Scholar
Srour, G., The independence relation in separably closed fields, this Journal, vol. 51 (1986), no. 3, pp. 715–725.Google Scholar
van den Dries, L., Model theory of valued fields lecture notes, available on the author’s homepage, 2004.Google Scholar
Wood, C., Notes on the stability of separably closed fields, this Journal, vol. 44 (1979), no. 3, pp. 412–416.Google Scholar