Minimal polynomial of an element in a polynomial quotient ring

Question

Let $p(z)$ be a (not necessarily irreducible) polynomial in $\mathbb{Q}[z]$. Is there a built-in function I can use for determining the minimal polynomial of an element $\overline{q(z)} \in \mathbb{Q}[z]/(p(z))$, i.e. finding the smallest degree nonzero monic polynomial $Q(z) \in \mathbb{Q}[z]$ for which $p(z) ~|~ Q(q(z))$?

MinimalPolynomial works, but only when $p(z)$ is irreducible and thus $\mathbb{Q}[z] / (p(z))$ determines a bona fide algebraic number field.

Answer 1

A specific example would help. I'll supply one though I do not know if it is quite what you have in mind.

We will use the extension below.

p = (z^2 - 3)*(z^3 + z^2 + 5*z + 4);

Now suppose we have an element in the extension ring. I picked the one below.

q = z^5 + z^4 - 7*z^2 + 2*z - 4;

The question is how to find a smallest polynomial that is satisfied by q. A way to do this is to set q to a new variable w and take the resultant with p, eliminating x.

Resultant[q - w, p, z]

(* Out[221]= (-107 + 32 w + w^2) (58148 + 4755 w - 75 w^2 + w^3) *)

Now we have a problem: which factor to use? Well, that depends on which factor in p we choose for z. So I guess Out[221] is the best we can do, in the absence of working over an irreducible polynomial extension.