Defining k in G k 1997

Abstract:  naming finitely many elements of the group as parameters) satisfied by a set of elements of the group such that the field operations on this set are again definable by first order formulas. In this way the field "lives" inside the group. Our construction yields just such a realization of the field in the group. We assume that the reader is familiar with the basic theory of linear algebraic groups. Most of what is needed here can be found in [Bo] or [BT1]. Throughout this note, k is a field, K ' k its algebraic closure, and G is a reductive k-group. Even though from the model theoretic point of view the question of definability becomes The second and third authors acknowledge the support of DFG grants. somewhat trivial if k is finite, we still think it worthwhile having a uniform treatment for all fields. A reductive group is always assumed to be connected. As usual