$(D,+,.)$ is finite division ring with characteristic $p$ - prime number. Prove that maping $f: D \rightarrow D, x \mapsto x^p $ is automorphism of the ring D.
Proof looks easier for $ ({\displaystyle \mathbb {Z}_p,+,.)}$ but how to make it general? Are there some differences? I know that I have to show that mapping is ring homomorphism and bijective.
Thank you for any help!