Pseudo algebraically closed fields over rings
type: Item
instance of: scholarly article
title: Pseudo algebraically closed fields over rings
zbMATH Open document ID: 0802.12007
DOI: 10.1007/BF02773673
author: Moshe Jarden
author: Aharon Razon
published in: Israel Journal of Mathematics
publication date: 1994-12-15T00:00:00Z
review text: 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 \(\mathbb Q\). \textit{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 \(V(\widetilde {\mathbb Q})\).
Mathematics Subject Classification ID: 12J12
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14A10
zbMATH DE Number: 590934
zbMATH Keywords: pseudo algebraically closed fields
zbMATH Keywords: absolute Galois group
zbMATH Keywords: Haar measure
zbMATH Keywords: affine absolutely irreducible variety
zbMATH Keywords: PAC field
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590934