Prove that each element in a finite field with characteristic $p$ is a $p$-th power for exactly one element in that field.

Question

I want to prove that for each element $x$ in $\mathbb{F}_q$ there exists a unique element $y \in \mathbb{F}_q$ such that $x ^p = y$, with $p$ the characteristic of the given field.

$\forall x \in \mathbb{F}_q: x^q = x$, with $q = p^h, p$ prime, $h \ge 1$. So we find $x^p = x^{1/h}$. Since the function $f:x\mapsto x^{1/h}$ is strictly monotonous, we can conclude that the property holds for $y = x^{1/h}$.

Is this a correct proof?