Suppose $a(T)$, $b(T) \in \mathbb{F}_p [T]$ are monic irreducible polynomials of the same degree $d$. The finite fields $\mathbb{F}_p [T] / (a(T))$ and $\mathbb{F}_p [T]/(b(T))$ are isomorphic, so there must exist some degree $d-1$ polynomial $q(T) \in \mathbb{F}_p [T]$ such that $b(T)$ divides $a(q(T))$.
Given $a(T)$ and $b(T)$, how can I construct such a $q(T)$ explicitly/concretely? (without, say, invoking uniqueness-up-to-isomorphism of finite fields) How many such $q(T)$ are there? (there should be $d$, since each such $q(T)$ gives an automorphism of the finite field?)