Additive inverses

Question

I have the problem of showing that for a field $F$ of characteristic $p$ and $n \in \mathbb{N}$, then the following set produces a subfield of $F$

$\{a \in $F$ : a^{p^n} = a$}

I have proved the other properties except am having trouble showing the existence of an additive inverse.

Any hints would be appreciated.

Answer 1

When $p=2$, $ch(F)=2$, so $2a=0$, hence $a=-a$. So $(-a)^{2^{n}}=a^{2^{n}}=a=-a$.