Equality in field with characteristic p

Question

Assume that $F$ is a field with characteristic $p$. How I can prove that in this field if we have $x^p=y^p$ then this implies that $x=y$ for any $x,y\in F$.

Answer 1

In characteristic $p$, $x^p-y^p=(x-y)^p$. So $x^p=y^p$ implies $(x-y)^p=0$ implies $x-y=0$ implies $x=y$.