Article contents
THE ELEMENTARY THEORY OF LARGE FIELDS OF TOTALLY $\mathfrak{S}$ -ADIC NUMBERS
Published online by Cambridge University Press: 23 April 2015
Abstract
We analyze the elementary theory of certain fields $K^{\mathfrak{S}}(\boldsymbol{\unicode[STIX]{x1D70E}})$ of totally $\mathfrak{S}$ -adic algebraic numbers that were introduced and studied by Geyer and Jarden and by Haran, Jarden, and Pop. In particular, we provide an axiomatization of these theories and prove their decidability, thereby giving a common generalization of classical decidability results of Jarden and Kiehne, Fried, Haran, and Völklein, and Ershov.
Keywords
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 1 , February 2017 , pp. 121 - 154
- Copyright
- © Cambridge University Press 2015
References
Basarab, S. A., The absolute Galois group of a pseudo real closed field with finitely many orders, J. Pure Appl. Algebra
38 (1985), 1–18.CrossRefGoogle Scholar
Chatzidakis, Z., Model theory of profinite groups, PhD thesis, Yale University (1984).Google Scholar
Chatzidakis, Z., Model theory of profinite groups having the Iwasawa property, Illinois J. Math.
42(1) (1998), 70–96.CrossRefGoogle Scholar
Chatzidakis, Z., Properties of forking in 𝜔-free pseudo-algebraically closed fields, J. Symbolic Logic
67(3) (2002), 957–996.CrossRefGoogle Scholar
Cherlin, G., van den Dries, L. and Macintyre, A., Decidability and undecidability theorems for PAC-fields, Bull. Amer. Math. Soc.
4(1) (1981), 101–104.CrossRefGoogle Scholar
Cherlin, G., van den Dries, L. and Macintyre, A., The elementary theory of regularly closed fields, Manuscript (1982).Google Scholar
Ershov, Y.,
PC
p
-fields with universal Galois group, Siberian Adv. Math.
1(4) (1991), 1–26.Google Scholar
Ershov, Y., Fields with continuous local elementary properties II, Algebra Logic
34(3) (1995), 140–146.CrossRefGoogle Scholar
Fehm, A., Decidability of large fields of algebraic numbers, PhD thesis, Tel Aviv University (2010).Google Scholar
Fehm, A., Elementary geometric local–global principles for fields, Ann. Pure Appl. Logic
164(10) (2013), 989–1008.CrossRefGoogle Scholar
Fried, M. D., Haran, D. and Völklein, H., Real Hilbertianity and the field of totally real numbers, in Arithmetic Geometry (ed. Childress, N. and Jones, J. W.), Contemporary Mathematics, Volume 174, pp. 1–34 (American Mathematical Society, 1994).CrossRefGoogle Scholar
Frohn, N., Model theory of absolute Galois groups, PhD thesis, University of Freiburg (Breisgau) (2010).Google Scholar
Geyer, W.-D. and Jarden, M., PSC Galois extensions of Hilbertian fields, Math. Nachr.
236(1) (2002), 119–160.3.0.CO;2-U>CrossRefGoogle Scholar
Haran, D., The undecidability of real closed fields, Manuscripta Math.
49 (1984), 91–108.CrossRefGoogle Scholar
Haran, D., Jarden, M. and Pop, F., The absolute Galois group of the field of totally S-adic numbers, Nagoya Math. J.
194 (2009), 91–147.CrossRefGoogle Scholar
Haran, D., Jarden, M. and Pop, F., The absolute Galois group of subfields of the field of totally S-adic numbers, Funct. Approx. Comment. Math.
46(2) (2012), 205–223.CrossRefGoogle Scholar
Jarden, M. and Kiehne, U., The elementary theory of algebraic fields of finite corank, Invent. Math.
30(3) (1975), 275–294.CrossRefGoogle Scholar
Jarden, M. and Ritter, J., On the characterization of local fields by their absolute Galois groups, J. Number Theory
11(1) (1979), 1–13.CrossRefGoogle Scholar
Jarden, M. and Razon, A., Rumely’s local global principle for algebraic PSC fields over rings, Trans. Amer. Math. Soc.
350(1) (1998), 55–85.CrossRefGoogle Scholar
Koenigsmann, J., From p-rigid elements to valuations (with a Galois-characterization of p-adic fields), J. Reine Angew. Math.
465 (1995), 165–182.Google Scholar
Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of Number Fields, second edn (Springer, 2008).CrossRefGoogle Scholar
Pop, F., Galoissche Kennzeichnung p-adisch abgeschlossener Körper, PhD Thesis, Heidelberg (1986).Google Scholar
Rumely, R. S., Undecidability and definability for the theory of global fields, Trans. Amer. Math. Soc.
262(1) (1980), 195–217.CrossRefGoogle Scholar
Shlapentokh, A., Hilbert’s Tenth Problem: Diophantine Classes and Extensions to Global Fields (Cambridge University Press, 2007).Google Scholar
- 1
- Cited by