# Remarks on automorphism groups of pseudo-finite fields

Zoé Chatzidakis, CNRS - ENS

Joint work with Özlem 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 its property. A pseudo-finite field is a field which is perfect, pseudo-algebraically closed, and with absolute Galois group isomorphic to $\hat{\mathbb Z}$.

Let $F$ be a pseudo-finite field, $F_0$ an elementary substructure, and ${\rm Aut}(F/F_0)$. We are interested in finite sections of ${\rm Aut}(F/F_0)$. In other words, let $A$ be a subfield of $F$ containing $F_0$, and let $B$ be the relative algebraic closure of $A$ inside $F$. Then $B={\rm acl}(A)$, and any element of ${\rm Aut}(B/A)$ is an elementary (partial) map of $F$. So the question is:

What are the possibilities for ${\rm Aut}(B/A)$?

We give precise answers to this question, thus extending previous results by Beyarslan and Hrushovski. The main result is:

**Theorem**. *Let $F$ be a pseudo-finite field, and $A$ a subfield of $F$ containing an elementary substructure $F_0$ of $F$, $B$ the relative algebraic closure of $A$ in $F$, and $G={\rm Aut}(B/A)$. If a prime $p$ divides $\#G$, then ${\rm char}(F_0)\neq p$, $F_0$ contains all primitive $p^n$-th roots of $1$ and $G$ is abelian.*