Every finitely generated regular field extension has a stable transcendence base
A field \(K\) is called stable if every finitely generated separable field extension \(F\) of \(K\) with \(K\) algebraically closed in \(F\) has a transcendence base \(x_1,\dots,x_n\) such that \(F\) is separable over \(K(x_1, \dots,x_n)\) and \(K\) is algebraically closed in the Galois hull \(\widehat F\) of \(F\) over \(K(x_1, \dots,x_n)\). In this paper it is proved that every field is stable. This generalizes results from [\textit{M. Fried} and \textit{M. Jarden}, Am. J. Math. 100, 653-666 (1976; Zbl 0413.12023), \textit{W.-D. Geyer} and \textit{M. Jarden}, Geom. Dedicata 29, 335-375 (1989; Zbl 0703.12006) and \textit{W.-D. Geyer}, Isr. J. Math. 68, 102-108 (1989; Zbl 0703.12007)]. The author applies the theorem for constructing fields \(K\) for which every absolutely irreducible variety has a least one \(K\)-rational point with some additional properties.
scientific article; zbMATH DE number 1251449
scientific article