$p^{th}$ roots of a field with characteristic $p$

We are looking for a root of $x^p-\alpha$; the formal derivative of this polynomial is zero, which means that $x^p-\alpha$ has repeated roots.

Indeed, if $K$ is an extension of $F$ where the polynomial has a root $\beta$, we have $$ (x-\beta)^p=x^p-\beta^p=x^p-\alpha $$ which shows the root is unique.

For a finite field $F$, the map $$ \alpha\mapsto\alpha^p $$ is a field homomorphism, so it is injective. Finiteness yields surjectivity.

If $a^p = b^p$ then $a^p-b^p = (a-b)^p=0$, and since you are in a field this implies $a=b$. This shows that for a field of characteristic $p$ the map $a \to a^p$ is always injective, and an injective map from a finite set to itself is automatically bijective.