Jordan–Chevalley decomposition

Question

I'm trying to understand the proof of it in Humphreys(Humphreys 1972, Prop. 4.2, p. 17). And I've not got over which field we are working. The characteristic polynomial may not have roots in the initial field, but he considers the endomorphisms of kind $ (x - a_i * id) $, where x is the endomorphism that we want to decompose, and $a_i$ is one of its roots.