$K$ field of characteristic $p$, where $p$ is prime

Question

Is there a way I can prove $(a+b)^{p^n} = a^{p^n} + b^{p^n}$, where int $n \ge 0$, $a,b\in K$ and $p$ prime the characteristic of $K$?

Can I use the binomial theorem and proof by induction? Both or just one?

Answer 1

You will want both, the binomial theorem says $p\big| {p\choose k}$ when $1\le k\le p-1$ so that

$$(a+b)^p\equiv a^p+b^p\mod p$$

Then using induction you see that

$$(a+b)^{p^n} = \big((a+b)^{p^{n-1}}\big)^p = (a^{p^{n-1}}+b^{p^{n-1}})^p = a^{p^n}+b^{p^n}.$$