) If $R$ is integral domaine If $char(R)=p$, then $(x +y)^p$ $=$ $ x^p +y^p$ and that $(x+y)^{p^n} = x^{p^n} +y^{p^n}$. (Hint: Use the fact that$ p |\binom {p} {k})$
I know that $R $ is integral dimain $x,y\in R $ if $xy=0$ then $x=0$ or $y=0$ $char(R)=p$, then $px=0$ $$ $$ But I don't know how use it , Can anyone help me in this problem ? Thanks .
Use the fact that $ \binom{p}{k} $ is divisible by $p$ for $k=1,\dotsc,p-1$ if $p$ is prime. Indeed, if $p\not \mid \binom{p}{k}$, then since $p\mid\binom{p}{k}k!$, it follows that $p\mid k!$ so that $p\mid i$ for some $i\leq k<p$ which is a contradiction.
Now use the binomial theorem together with this fact.