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Pseudo algebraically closed fields over rings. (English) Zbl 0802.12007
A field \(M\) is said to be pseudo algebraically closed (PAC) if every absolutely irreducible variety \(V\) defined over \(M\) has an \(M\)-rational point. This property was formulated by J. Ax, and later Frey called it PAC. Denote by \(\widetilde {\mathbb Q}\) the algebraic closure of \(\mathbb Q\), by \(G(\mathbb Q)\) the absolute Galois group of \(\mathbb Q\). M. Jarden [Trans. Am. Math. Soc. 164, 67–91 (1972; Zbl 0235.12104)] proved that if \(e\) is a positive integer, then \(\widetilde {\mathbb Q} (\sigma)\) is PAC for almost all \(\sigma\in G(\mathbb Q)^ e\) in the sense of the Haar measure of \(G(\mathbb Q)^ e\). Later on more examples of algebraic extensions of \(\mathbb Q\) which are PAC were given.
In the present paper the authors introduce a stronger property. Let \(O\) be a subset of a field \(M\). The field \(M\) is said to be PAC over \(O\) if for every affine absolutely irreducible variety \(V\) of dimension \(n\geq 0\) and for each dominating separable rational map \(\varphi: V\to A^ r\) over \(M\) there exists \(a\in V(M)\) such that \(\varphi(a)\in O^ r\). Each PAC field is PAC over itself. If \(M\) is a separably closed field and \(O\) is an infinite subring, then \(M\) is PAC over \(O\). It is proved that almost all fields \(\mathbb Q(\sigma)\) are PAC over \(\mathbb Z\). Moreover, almost all fields \(\widetilde {\mathbb Q}(\sigma)\) have a ‘density property’: For each valuation \(w\) of \(\mathbb Q\) and absolutely irreducible variety \(V\) defined over \(\widetilde {\mathbb Q}(\sigma)\) the set \(V(\widetilde {\mathbb Q} (\sigma))\) is \(w\)-dense in \(V(\widetilde {\mathbb Q})\).
Reviewer: G.Pestov (Tomsk)

12J12 Formally \(p\)-adic fields
14E05 Rational and birational maps
14A10 Varieties and morphisms
Full Text: DOI
[1] J. Ax,The elementary theory of finite fields, Annals of Mathematics88 (1968), 239–271. · Zbl 0195.05701 · doi:10.2307/1970573
[2] J.W.S. Cassels and A. Fröhlich,Algebraic Number Theory, Academic Press, London, 1967.
[3] G. Faltings,Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae73 (1983), 349–366. · Zbl 0588.14026 · doi:10.1007/BF01388432
[4] G. Frey,Pseudo algebraically closed fields with non-archimedian real valuations, Journal of Algebra26 (1973), 202–207. · Zbl 0264.12105 · doi:10.1016/0021-8693(73)90020-3
[5] M. Fried and M. Jarden, Diophantine properties of subfields of \(\tilde{\mathbb{Q}}\) , American Journal of Mathematics100 (1978), 653–666. · Zbl 0413.12023 · doi:10.2307/2373846
[6] M. D. Fried and M. Jarden,Field Arithmetic, Ergebnisse der Mathematik (3)11, Springer, Heidelberg, 1986.
[7] M. Fried and H. Völklein,The inverse Galois problem and rational points on moduli spaces, Mathematische Annalen290 (1991), 771–800. · Zbl 0763.12004 · doi:10.1007/BF01459271
[8] W.-D. Geyer and M. Jarden,On stable fields in positive characteristic, Geometria Dedicata29 (1989), 335–375. · Zbl 0703.12006 · doi:10.1007/BF00572450
[9] 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. · Zbl 0652.12010 · doi:10.1515/crll.1988.383.147
[10] 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. · Zbl 0595.12013
[11] D. Harbater,Galois coverings of the arithmetic line, Lecture Notes in Mathematics1240, Springer, Berlin, 1987, pp. 165–195. · Zbl 0627.12015
[12] M. Jarden,Elementary statements over large algebraic fields, Transactions of AMS164 (1972), 67–91. · Zbl 0235.12104 · doi:10.1090/S0002-9947-1972-0302651-9
[13] 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. · Zbl 0737.12001
[14] M. Jarden,The inverse Galois problem over formal power series fields, Israel Journal of Mathematics85 (1994), 263–275. · Zbl 0794.12004 · doi:10.1007/BF02758644
[15] M. Jarden and Peter Roquette,The Nullstellensatz over p-adically closed fields, Journal of the Mathematical Society of Japan32 (1980), 425–460. · Zbl 0446.12016 · doi:10.2969/jmsj/03230425
[16] S. Lang,Introduction to Algebraic Geometry, Interscience Publishers, New York, 1958. · Zbl 0095.15301
[17] S. Lang,Algebra, Addison-Wesley, Reading, 1970.
[18] S. Lang,Algebraic Number Theory, Addison-Wesley, Reading, 1970. · Zbl 0211.38404
[19] B. H. Matzat,Der Kenntnisstand in der Konstruktiven Galoischen Theorie, manuscript, Heidelberg, 1990.
[20] D. Mumford,The Red Book of Varieties and Schemes, Lecture Notes in Mathematics1358, Springer, Berlin, 1988. · Zbl 0658.14001
[21] F. Pop,Fields of totally \(\Sigma\)-adic numbers, manuscript, Heidelberg, 1992.
[22] A. Prestel,Pseudo real closed fields, inSet Theory and Model Theory, Lecture Notes in Math.872, Springer, Berlin, 1981, pp. 127–156.
[23] M. Rzedowski-Calderón and G. Villa-Salvador,Automorphisms of congruence function fields, Pacific Journal of Mathematics150 (1991), 167–178. · Zbl 0694.12011
[24] P. Samuel,Lectures on Old and New Results on Algebraic Curves, Tata Institute of Fundamental Research, Bombay, 1966. · Zbl 0165.24102
[25] J.-P. Serre,Topics in Galois Theory, Jones and Barlett, Boston, 1992. · Zbl 0746.12001
[26] J.-P. Serre,A Course in Arithmetic, Graduate Texts in Mathematics7, Springer, New York, 1973.
[27] I.R. Shafarevich,Basic algebraic Geometry, Grundlehren der mathematischen Wissenschaften213, Springer, Berlin, 1977.
[28] H. Völklein, Braid groups, Galois groups and cyclic covers of \(\mathbb{P}\) , manuscript, 1992.
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