On the compositum of algebraically closed subfields

On the Compositum of Algebraically Closed Subfields

1. S. Abhyankar, On the ramification of algebraic functions, Amer. J. Math. vol. 77 (1955) pp. 575-592. 2. C. Chevalley, On the theory of local rings, Ann. of Math. vol. 44 (1943), pp. 680-708. 3. -, Introduction to the theory of algebraic functions of one variable, New York, 1951. 4. S. MacLane and O. F. G. Schilling, Zero-dimensional branches of rank one on algebraic varieties, Ann. of Math. ...

On Pseudo Algebraically Closed Extensions of Fields

The notion of ‘Pseudo Algebraically Closed (PAC) extensions’ is a generalization of the classical notion of PAC fields. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure....

Pseudo Algebraically Closed Extensions

Deenability of Geometric Properties in Algebraically Closed Fields Deenability of Geometric Properties in Algebraically Closed Fields Deenability of Geometric Properties in Algebraically Closed Fields

We prove that there exists no sentence F of the language of rings with an extra binary predicat I satisfying the following property for every de nable set X C X is connected if and only if C X j F where I is interpreted by X We conjecture that the same result holds for the closed subsets of C We prove some results motivated by this conjecture

Imaginaries in pairs of algebraically closed fields

We consider the theory (ACFp)P of pairs F < K of algebraically closed fields of a given characteristic p. We exhibit a collection of additional sorts in which this theory has geometric elimination of imaginaries. The sorts are essentially of the form ∪a∈B(F )Va(K)/Ga(F ), where G, V,B are varieties over the prime field G a group scheme over B and V is a scheme over B (both with irreducible fibr...