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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)

##### MSC:
 12J12 Formally $$p$$-adic fields 14E05 Rational and birational maps 14A10 Varieties and morphisms
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