Let $G$ be a group scheme over a field $F$, and let $f:G\to G$ be a homomorphism. Written in my notes, I have the following statement:
To check surjectivity (on $F$-rational points), it suffices to show that the induced morphism $G_K\to G_K$ is surjective, for some finite extension $K/F$.
I would like a reference for this fact.
Remarks
- As such, this statement may be (very) wrong. Among the hypotheses that I have on $G$: it is a connected reductive algebraic group. However, the above is stated in this level of generality because I want to see what breaks down if we don't assume enough.
- Moreover, one might need further hypotheses on the morphism $f$.