The algebraic nature of the elementary theory of PRC fields
A field K is PAC (resp. PRC) if every nonempty absolutely irreducible variety V defined over K has a K-rational point (resp. if each such V has a K-rational point provided it has a simple \(\bar K\)-rational point for every real closure \(\bar K\) of K). - After \textit{M. Fried} and \textit{M. Jarden} [``Field arithmetic'', Ergebn. Math. Grenzgebiete, 3. Folge 11 (1986; Zbl 0625.12001)], the elementary theory of algebraic PAC fields determines the elementary theory of all PAC fields (thus a sentence \(\theta\) is true in each field of characteristic 0 if it is true in each PAC field which is algebraic over \({\mathbb{Q}}).\) The scope of this paper is to prove an analogous result for PRC and maximal PRC fields. Also is proved that the absolute Galois group of a maximal PRC field is a free product of groups of order 2 in the category of pro-2 groups and conversely, each such group is isomorphic to the absolute Galois group of a maximal PRC field.
elementary theory of algebraic PAC fields
scientific article; zbMATH DE number 4105089