Independence, measure and pseudofinite fields

2006 ◽  
Vol 12 (2) ◽  
pp. 271-306
Author(s):  
Ivan Tomašić
Keyword(s):  
Pseudofinite Fields

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

From Stability to Simplicity

1998 ◽  
Vol 4 (1) ◽  
pp. 17-36 ◽  
Author(s):  
Byunghan Kim ◽  
Anand Pillay
Keyword(s):  
Recent Work ◽  
Stability Theory ◽  
Simple Theories ◽  
First Order ◽  
Closed Fields ◽  
Exhaustive Study ◽  
Pseudofinite Fields ◽  
Algebraically Closed Fields ◽  
Major Progress ◽  
Made In

§1. Introduction. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular algebraic structures which have been studied recently (pseudofinite fields, algebraically closed fields with a generic automorphism, smoothly approximable structures) turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2λ many models in any cardinal λ ≥ ω1 + |T| (proved by Shelah). (Note that for λ ≥ |T| 2λ is the maximum possible number of models of cardinality λ.)

scholarly journals Pseudofinite fields, procyclic fields and model-completion

1976 ◽  
Vol 62 (2) ◽  
pp. 305-309 ◽  
Author(s):  
Allan Adler ◽  
Catarina Kiefe
Keyword(s):  
Pseudofinite Fields ◽  
Model Completion

scholarly journals Exponential sums in pseudofinite fields and applications

2004 ◽  
Vol 48 (4) ◽  
pp. 1235-1257 ◽  
Author(s):  
Ivan Tomašić
Keyword(s):  
Exponential Sums ◽  
Pseudofinite Fields

scholarly journals Coding complete theories in Galois groups

2008 ◽  
Vol 73 (2) ◽  
pp. 474-491
Author(s):  
James Gray
Keyword(s):  
Galois Groups ◽  
Vietoris Topology ◽  
Closed Fields ◽  
Prime Fields ◽  
Complete Theories ◽  
Pseudofinite Fields ◽  
Algebraically Closed Fields

AbstractIn this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

Random hypergraphs in pseudofinite fields

2009 ◽  
Vol 9 (1) ◽  
pp. 29-47 ◽  
Author(s):  
Özlem Beyarslan
Keyword(s):  
Function Field ◽  
Galois Groups ◽  
Alternating Groups ◽  
Generic Polynomials ◽  
Pseudofinite Fields ◽  
Random Hypergraphs

AbstractWe prove thatn-hypergraphs can be interpreted in e-free perfect PAC fields in particular in pseudofinite fields. We use methods of function field arithmetic, more precisely we construct generic polynomials with alternating groups as Galois groups over a function field.

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