$e^{2\pi i/k}\mapsto e^{2\pi i m/k}$ automorphism on the cyclotomic field $\mathbb{Q}(e^{2\pi i/k})$

Question

I am currently reading a proof in which they mention $e^{2\pi i/k}\mapsto e^{2\pi i m/k}$ as an automorphism on the cyclotomic field $\mathbb{Q}(e^{2\pi i/k})$. I know next to nothing about cyclotomic fields and the books that i have been able to find also just make this claim without proof.

My question is how would you prove that this is an automorphism on that field?