Help with a basic question on Finite Field characteristic

Question

Let $F$ be a field of characteristic $p > 0$.

Show that $(\alpha + \beta)^{p^m} = \alpha^{p^m}+\beta^{p^m}$, for all $\alpha,\beta \in F$ and $m > 0$.

I am stuck on this question; can anybody help?

Answer 1

Hint:

Prove $\;\alpha^p+\beta^p=(\alpha+\beta)^p$ first (use the binomial formula), then use induction on $m$, noting that $$\Bigl(\alpha^{p^m}+\beta^{p^m}\Bigr)^p=(\alpha+\beta)^{p^{m+1}}.$$