Pseudo algebraically closed fields over rings
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 \(V(\widetilde {\mathbb Q})\).
scientific article; zbMATH DE number 590934
scientific article