Prove that solution of equation $x^p=a$ exists, where $a$ is a fixed element in a finite field $F$ and $charF=p.$

Question

I have to prove that solution of equation $x^p=a$ exists and it's unique, where $a$ is a fixed element in a finite field $(F,+,\cdot,0,1)$ and $charF=p.$ I know how to prove that $p$ is a prime number and using binomial formula it is easy to prove $(x-y)^p=x^p-y^p,$ so the solution is unique. I don't know how to prove that the solution exists. For $a=0,$ $x=0$ and for $a=1,$ $x=1.$ But what can be a solution if $a\neq0,$ $a\neq1?$

Answer 1

Define $\varphi: F\to F$ by $x\to x^p$. It is a field homomorphism, hence injective. Since it is an injective map from a finite set to itself it must be surjective as well.