DefiningkinG(k)

We show how the field of definitionkof ak-isotropic absolutely almost simplek-groupG“lives” in the groupG(k) ofk-rational points. The construction which is inspired by the fundamental work of Borel-Tits is as follows: We choose an element inside the center of the unipotent radical of a minimal parabolick-subgroupP; the orbit under the action of the centerZof a Levik-subgroup ofPgenerates a one-dimensional vector space which then carries the additive field structure in a natural way. The multiplicative structure is induced by the action ofZ. IfGisk-simple, our construction yields a finite extensionlofk. As an immediate consequence we obtain an answer to a question of Borovik–Nesin under the additional assumption thatGisk-isotropic: T . IfGis ak-simplek-isotropic group such thatG(k) has finite Morley rank, thenkis either algebraically closed or real closed. IfGis absolutely simplek-isotropic, thenkis algebraically closed.