We define the Frobenius Homomorphism as that:
Let $F$ be a field of characteristic $p\gt 0$. Then we call the Frobenius homomorphism this map: $$\phi:F\to F, \phi(x)=x^p$$
I have the following questions:
- When this homomorphism is in fact an endomorphism or an automorphism?
- Why $x^p=0$ implies $x=0$ (I think it's true in a finite field $F$, but I don't know why)
I would appreciate it so much if someone could help me with this. I think it's a common doubt for beginners because I've already read some books without any clarification on this topic.