Regularly \(T\)-closed fields. (Q2715535)

\textit{B. Heinemann} and \textit{A. Prestel} were the first authors who treated in a systematic way local-global principles for the existence of rational points on absolutely irreducible varieties [Can. Math. Soc. Proc. 4, 297--336 (1984; Zbl 0555.12008)]. In the paper under review a further concept of a local-global principle is introduced and results of Heinemann and Prestel are generalized.NEWLINENEWLINELet \(T\) be a first order theory in the language of fields. It is assumed that \(T\) extends the theory of fields. If \(K\) is a field then an algebraic extension \(F\) of \(K\) is called a \(T\)-closure of \(K,\) if \(F\) is the relative separable closure of \(K\) in some extension field \(E\) with \(E\models T.\) The set of all \(T\)-closures of \(K\) contained in some fixed algebraic closure \(\widetilde{K}\) of \(K\) is denoted by \(X_K^T.\) If \(T_1,\dots,T_n\) are theories of fields then the theory of all fields that are models of some \(T_i\) is denoted by \(T.\) Then for a field \(K\), \(X_K^T=\bigcup_{i=1}^{i=n}X_K^{T_i}.\) Introduce in \(X_K^T\) the coarsest topology such that for all non-constant separable polynomials \(f\in K[X]\) the sets NEWLINE\[NEWLINEH^+(f)=\{F\in X_K^T\mid F\models \exists x f(x)=0\},NEWLINE\]NEWLINE NEWLINE\[NEWLINEH^-(f)=\{F\in X_K^T\mid F\models \neg\exists x f(x)=0\}NEWLINE\]NEWLINE are open. The set \(X_K^T\) is quasi-compact for each field \(K.\) It is shown that for every existential formula \(\phi\) in the language of fields with parameters from the field \(K\) the set NEWLINE\[NEWLINE\{F\in X_K^T\mid F\models\phi \}NEWLINE\]NEWLINE is open. The set \(X_K^T\) is quasi-compact for each field \(K.\)NEWLINENEWLINEA field \(K\) is called regularly \(T\)-closed if every absolutely irreducible variety \(V\subset {\widetilde{K}}^n\) defined over \(K\) which has a regular point in each \(T\)-closure of \(K\) also has a \(K\)-rational point. A variety here is understood to be an affine variety which is embedded in some affine space over an algebraically closed field. It is a well-known fact that pseudo algebraically closed, pseudo real closed and pseudo \(p\)-adically closed fields can be characterized in terms of affine plane curves. The author generalizes this result to regularly \(T\)-closed fields.NEWLINENEWLINEA field \(K\) is said to satisfy the curve condition if every absolutely irreducible affine plane curve \(C\) defined over \(K\) which has an \(F\)-rational regular point for each \(F\in X_K^T\) also admits a \(K\)-rational point. It is proved that a finite field \(K\) satisfies the curve condition if and only if \(K\in X_K^T.\) The class of regularly \(T\)-closed fields is elementary and inductive.NEWLINENEWLINEAn approximation theorem for regularly closed fields is proved which represents a generalization of a result of Heinemann and Prestel (loc. cit., Theorem 1.9).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00034].