If $R$ is an integral domain with char $p$ where $p>0$ and $f:R\to R$ where $f(x)=x^p$
How would one go about showing addition is preserved? e.g. $f(a+b)=f(a)+f(b)$? Multiplication is obvious. So far for addition I have:
$f(a+b)=(a+b)^p=\sum\limits_{r=0}^p {p\choose r}a^rb^{p-r}=a^p+\sum\limits_{r=1}^{p-1} {p\choose r}a^rb^{p-r}+b^p=f(a)+\sum\limits_{r=1}^{p-1} {p\choose r}a^rb^{p-r}+f(b)$
So i'm currently having difficulty proving $\sum\limits_{r=1}^{p-1} {p\choose r}a^rb^{p-r}=0$
Also can anyone think of an example such that $f$ isn't surjective?