Abstract
We prove that for almost allσ ∈G ℚ the field\(\tilde{\mathbb{Q}}\) has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→\(\tilde{\mathbb{Q}}\) there exists a point a ∈\(\mathbb{A}\) such thatϕ(a) ∈ ℤr. We then say that\(\tilde{\mathbb{Q}}\) is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields\(\tilde{\mathbb{Q}}\) other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Ax,The elementary theory of finite fields, Annals of Mathematics88 (1968), 239–271.
J.W.S. Cassels and A. Fröhlich,Algebraic Number Theory, Academic Press, London, 1967.
G. Faltings,Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae73 (1983), 349–366.
G. Frey,Pseudo algebraically closed fields with non-archimedian real valuations, Journal of Algebra26 (1973), 202–207.
M. Fried and M. Jarden, Diophantine properties of subfields of\(\tilde{\mathbb{Q}}\), American Journal of Mathematics100 (1978), 653–666.
M. D. Fried and M. Jarden,Field Arithmetic, Ergebnisse der Mathematik (3)11, Springer, Heidelberg, 1986.
M. Fried and H. Völklein,The inverse Galois problem and rational points on moduli spaces, Mathematische Annalen290 (1991), 771–800.
W.-D. Geyer and M. Jarden,On stable fields in positive characteristic, Geometria Dedicata29 (1989), 335–375.
D. Haran and M. Jarden,The absolute Galois group of a pseudo p-adically closed field, Journal für die reine und angewandte Mathematik383 (1988), 147–206.
D. Haran and M. Jarden,The absolute Galois group of a pseudo real closed field, Annali della Scuola Normale Superiore — Pisa, Serie IV,12 (1985), 449–489.
D. Harbater,Galois coverings of the arithmetic line, Lecture Notes in Mathematics1240, Springer, Berlin, 1987, pp. 165–195.
M. Jarden,Elementary statements over large algebraic fields, Transactions of AMS164 (1972), 67–91.
M. Jarden,Intersection of local algebraic extensions of a Hilbertian field (edited by A. Barlotti et al.), NATO ASI Series C333, Kluwer, Dordrecht, 1991, pp. 343–405.
M. Jarden,The inverse Galois problem over formal power series fields, Israel Journal of Mathematics85 (1994), 263–275.
M. Jarden and Peter Roquette,The Nullstellensatz over p-adically closed fields, Journal of the Mathematical Society of Japan32 (1980), 425–460.
S. Lang,Introduction to Algebraic Geometry, Interscience Publishers, New York, 1958.
S. Lang,Algebra, Addison-Wesley, Reading, 1970.
S. Lang,Algebraic Number Theory, Addison-Wesley, Reading, 1970.
B. H. Matzat,Der Kenntnisstand in der Konstruktiven Galoischen Theorie, manuscript, Heidelberg, 1990.
D. Mumford,The Red Book of Varieties and Schemes, Lecture Notes in Mathematics1358, Springer, Berlin, 1988.
F. Pop,Fields of totally Σ-adic numbers, manuscript, Heidelberg, 1992.
A. Prestel,Pseudo real closed fields, inSet Theory and Model Theory, Lecture Notes in Math.872, Springer, Berlin, 1981, pp. 127–156.
M. Rzedowski-Calderón and G. Villa-Salvador,Automorphisms of congruence function fields, Pacific Journal of Mathematics150 (1991), 167–178.
P. Samuel,Lectures on Old and New Results on Algebraic Curves, Tata Institute of Fundamental Research, Bombay, 1966.
J.-P. Serre,Topics in Galois Theory, Jones and Barlett, Boston, 1992.
J.-P. Serre,A Course in Arithmetic, Graduate Texts in Mathematics7, Springer, New York, 1973.
I.R. Shafarevich,Basic algebraic Geometry, Grundlehren der mathematischen Wissenschaften213, Springer, Berlin, 1977.
H. Völklein, Braid groups, Galois groups and cyclic covers of\(\mathbb{P}\), manuscript, 1992.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jarden, M., Razon, A. Pseudo algebraically closed fields over rings. Israel J. Math. 86, 25–59 (1994). https://doi.org/10.1007/BF02773673
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02773673