A field K is called pseudo-algebraically closed (PAC), if every absolutely irreducible variety defined over K has a K-rational point. This class of fields was introduced in 1968 by Ax on the way to his famous algebraic characterization of the pseudo-finite fields: a field is elementarily equivalent to an ultra-product of finite fields if and only if it is perfect, has free profinite absolute Galois group on a single generator, and is PAC. In the decades that followed, the PAC fields were an intensive object of study within model-theoretic algebra. Cherlin, van den Dries, and MacIntyre gave complete invariants for the first-order theory of a PAC field in terms of its characteristic, absolute numbers, and the 'co-theory' of its absolute Galois group, in a formalism they called 'co-logic'. This 'co-logic' approach was presented within first-order logic by Chatzidakis who, later on, showed how to connect model-theoretic properties of the theory of the absolute Galois group of a PAC field to the model-theoretic properties of the field itself. We will survey this work in detail and give some recent applications to classification-theoretic questions concerning PAC fields. Time-permitting, we will also talk about some work in progress and many open problems in this area.
Mon, 15/05/2017 - 14:00 to 16:00