It is well-known that a definable field of finite Morley rank has no proper definable group of automorphisms (a proof can be found for example on the book "Stable group" of Poizat). My question is: can be said something similar on the morphisms of the additive group $K^+$ that are morphisms of a definable proper infinite subgroup of $K^{\times}$. Clearly we are in a higly patological situation, indeed the field must be of characteristic $p$ (if not any additive morphism is a product by a scalar and therefore it must be the identity in our case) and a green field (since it must have a proper infinite definable multiplicative subgroup). Green fields of finite characteristic are unlikely to exists but I didn't find a proof of the non-existence of them.

It is well-known that a definable field of finite Morley rank has no proper definable group of automorphisms (a proof can be found for example in the book "Stable groups" of Poizat). My question is: can something similar be said about morphisms of the additive group $K^+$ that are morphisms of a definable proper infinite subgroup of $K^{\times}$? Clearly we are in a higly pathological situation, indeed the field must be of characteristic $p$ (if not any additive morphism is a product by a scalar and therefore it must be the identity in our case) and a green field (since it must have a proper infinite definable multiplicative subgroup). Green fields of finite characteristic are unlikely to exist but I didn't find a proof of the non-existence of them.