Let p be a prime. Show that for any two elements $a, b$ in a field of characteristic $p$, $(a+b)^{p^k} = a^{p^k} +b^{p^k}$ for any $k \geq 0$
My intuition tells me that I should proceed by saying that $p^k$ is the order of the field and use that to prove the result, but i'm not sure that is true.
Use the binomial theorem and induction: since $x^{n^k}$ is just $x$ raised to the $n$ a total of $k$ times, you see that it is enough to show that $(a+b)^p=a^p+b^p\mod p$. And this follows from the binomial theorem and the fact that $p \big| {p\choose k}$ when $1\le k\le p-1$.