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


Zbl 0235.12104
Full Text: DOI


[1] Ax, J., The elementary theory of finite fields, Annals of Mathematics, 88, 239-271 (1968) · Zbl 0195.05701
[2] Cassels, J. W.S.; Fröhlich, A., Algebraic Number Theory (1967), London: Academic Press, London · Zbl 0153.07403
[3] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae, 73, 349-366 (1983) · Zbl 0588.14026
[4] Frey, G., Pseudo algebraically closed fields with non-archimedian real valuations, Journal of Algebra, 26, 202-207 (1973) · Zbl 0264.12105
[5] Fried, M.; Jarden, M., Diophantine properties of subfields of \(\tilde{\mathbb{Q}} \), American Journal of Mathematics, 100, 653-666 (1978) · Zbl 0413.12023
[6] Fried, M. D.; Jarden, M., Field Arithmetic (1986), Heidelberg: Springer, Heidelberg · Zbl 0625.12001
[7] Fried, M.; Völklein, H., The inverse Galois problem and rational points on moduli spaces, Mathematische Annalen, 290, 771-800 (1991) · Zbl 0763.12004
[8] Geyer, W.-D.; Jarden, M., On stable fields in positive characteristic, Geometria Dedicata, 29, 335-375 (1989) · Zbl 0703.12006
[9] Haran, D.; Jarden, M., The absolute Galois group of a pseudo p-adically closed field, Journal für die reine und angewandte Mathematik, 383, 147-206 (1988) · Zbl 0652.12010
[10] Haran, D.; Jarden, M., The absolute Galois group of a pseudo real closed field, Annali della Scuola Normale Superiore — Pisa, 12, 449-489 (1985) · Zbl 0595.12013
[11] Harbater, D., Galois coverings of the arithmetic line, 165-195 (1987), Berlin: Springer, Berlin
[12] Jarden, M., Elementary statements over large algebraic fields, Transactions of AMS, 164, 67-91 (1972) · Zbl 0235.12104
[13] Jarden, M.; Barlotti, A., Intersection of local algebraic extensions of a Hilbertian field, 343-405 (1991), Dordrecht: Kluwer, Dordrecht · Zbl 0737.12001
[14] Jarden, M., The inverse Galois problem over formal power series fields, Israel Journal of Mathematics, 85, 263-275 (1994) · Zbl 0794.12004
[15] Jarden, M.; Roquette, Peter, The Nullstellensatz over p-adically closed fields, Journal of the Mathematical Society of Japan, 32, 425-460 (1980) · Zbl 0446.12016
[16] Lang, S., Introduction to Algebraic Geometry (1958), New York: Interscience Publishers, New York · Zbl 0095.15301
[17] Lang, S., Algebra (1970), Reading: Addison-Wesley, Reading · Zbl 0216.06001
[18] Lang, S., Algebraic Number Theory (1970), Reading: Addison-Wesley, Reading · Zbl 0211.38404
[19] B. H. Matzat,Der Kenntnisstand in der Konstruktiven Galoischen Theorie, manuscript, Heidelberg, 1990.
[20] Mumford, D., The Red Book of Varieties and Schemes (1988), Berlin: Springer, Berlin · Zbl 0658.14001
[21] Pop, F., Fields of totally Σ-adic numbers (1992), Heidelberg: manuscript, Heidelberg
[22] Prestel, A., Pseudo real closed fields, Set Theory and Model Theory, 127-156 (1981), Berlin: Springer, Berlin
[23] Rzedowski-Calderón, M.; Villa-Salvador, G., Automorphisms of congruence function fields, Pacific Journal of Mathematics, 150, 167-178 (1991) · Zbl 0694.12011
[24] Samuel, P., Lectures on Old and New Results on Algebraic Curves (1966), Bombay: Tata Institute of Fundamental Research, Bombay · Zbl 0165.24102
[25] Serre, J.-P., Topics in Galois Theory (1992), Boston: Jones and Barlett, Boston · Zbl 0746.12001
[26] Serre, J.-P., A Course in Arithmetic (1973), New York: Springer, New York · Zbl 0256.12001
[27] Shafarevich, I. R., Basic algebraic Geometry (1977), Berlin: Springer, Berlin · Zbl 0362.14001
[28] H. Völklein, Braid groups, Galois groups and cyclic covers of \(\mathbb{P} \) , manuscript, 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.