Model theory of finite fields and pseudo-finite fields (Q1377639)

This article gives a survey of the main results obtained to date on finite fields and pseudo-finite fields. In his famous paper ``The elementary theory of finite fields'' [Ann. Math. (2) 88, 239-271 (1968; Zbl 0195.05701)], \textit{J. Ax} has shown that the set \(T_f\) of sentences true in all finite fields is decidable. The infinite models of \(T_f\) are called `pseudo-finite fields'. The survey under consideration starts by explaining Ax's results. It then discusses further improvements concerning the decision procedures, Galois stratification, definability, Zeta function of a formula, Haar measure on the absolute Galois group and related results.