Zo´e Chatzidakis (CNRS - ENS), Remarks on automorphism groups of pseudo-finite fields. ¨ Joint work with Ozlem Beyarslan. Recall that a pseudo-finite field is an infinite model of the theory of finite fields. They were extensively studied last century, starting with the work of Ax in the 1960’s which described their theory and properties. A pseudo-finite field is a field which is perfect, pseudo-algebraically ˆ closed, and with absolute Galois group isomorphic to Z. Let F be a pseudo-finite field, F0 an elementary substructure, and consider Aut(F/F0 ). We are interested in the finite sections of Aut(F/F0 ). In other words, let A be a subfield of F containing F0 , and let B be the relative algebraic closure of A inside F . Then B = acl(A), and any element of Aut(B/A) defines an elementary (partial) map of F . So the question is: What are the possibilities for Aut(B/A)? ¨ Beyarslan and We give precise answers to this question, thus extending previous results by O. E. Hrushovski. The main result is: Theorem. Let F be a pseudo-finite field, and A a subfield of F containing an elementary substructure F0 of F , B the relative algebraic closure of A in F , and G = Aut(B/A). If a prime p divides #G, then char(F0 ) 6= p, F0 contains all primitive pn -th roots of unity and G is abelian.

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