Let $f \in F_p[x] / p(x)$. I know for a fact that the inverse of $f$, $g$ exists in the field.

I'm looking for algorithms to find the inverse of $g$.

When can we find a $g(x) \in F_p[x]/p(x)$ so that $f(g(x)) =x$?

Additionally, if you know of any papers that talk about the existence of inverses that would also be useful.

Let $f \in F_p[x] / q(x)$. I know for a fact that the inverse of $f$, $g$ exists in the field. Taking the regular inverse is easy, but I'm looking for the compositional inverse.

I'm looking for algorithms to find the inverse of $f$.

When can we find a $g(x) \in F_p[x]/q(x)$ so that $f(g(x)) =x$?

Additionally, if you know of any papers that talk about the existence of inverses that would also be useful.