If two polynomials divide each other in a field such as $\mathbb{R}$ or $\mathbb{Q}$, then they certainly have to differ by a constant: $g(x) = k f(x)$.
But what about a finite field? If polynomials divide each other, are they still constant multiples of each other by some $k \in F$ where $f$ is finite?
My logic is that if $f(x)\mid g(x)$ and $g(x)\mid f(x)$ in a finite field, then there exists polynomials $p(x)$ and $q(x)$ such that $g(x)p(x) = f(x)$ and $f(x)q(x) = g(x)$.
Now $f(x) q(x) p(x) = f(x)$ or $q(x) p(x) = 1$. But I am stuck here. I doubt that $q(x) p(x) = 1$ could alone imply that $q(x)$ and $p(x)$ are constants in a finite field since inverse polynomials can exist.