Let $R$ be a commutative unital ring of characteristic $p$, where $p$ is prime. The Frobenius map $x \mapsto x^p$ on $R$ is known to be a ring homomorphism (in particular, additive).
The only proof I know of this fact is to use binomial expansion and then $p \mid {p\choose k}$ for every $0<k<p$. But what are some other proofs? Are there "conceptual" reason for such a canonical morphism to exist?