I'm trying to understand the proof of Wedderburn's theorem which states that every finite division ring is a field. I'm following the proof given by Herstein in his book Algebra. Wedderburn's theorem is preceded by a preliminary lemma(see the picture below) whose proof I have a doubt about.
The passage that creates difficulties for me is the one in which the polynomial $$t^{p^n}-t$$ over $P(a)$ is evaluated in the map $\delta$. What are the profound reasons that justify this substitution? Is it always possible or only if the map commutes with the scalars on which the ring of polynomials is built? In the second case, why? Is it only a formal substitution? When this substitution is made, does one pass into the algebra of endomorphisms and therefore must I consider the second product as a composition between applications?
