I would really appreciate help in this question, or even a good direction for an answer. I have looked in the appropriate literature without much progress..
Q1:
Let $p(x)\in\mathbb{Q}[x]$ be the minimal polynomial of $\omega\in\mathbb{C}$, and let $\mathcal{R}$ be the quotient polynomial ring over the ideal created by $p(x)$ $$\mathcal{R}=\mathbb{Q}[x]/(p(x))$$ For a given polynomial $q(x)\in\mathcal{R}$, given $q(\omega)$, what is $q(x)$? is there a closed formula to find $q(x)$ in $\mathcal{R}$?
It is easy to see that $q(x)$ is uniquely determined by the value $q(\omega)$ (if $q_1(\omega)=q_2(\omega)$ their difference is the $0\in\mathcal{R}$ polynomial)
Example
Even for small rank polynomials I am not sure what to do, for example when $p(x)=x^4+1$ is the minimal polynomial of the 8th root of unity $exp(\frac{2\pi i}{8})$.
Naive attempt
The Lagrange Polynomial could provide the correct polynomial, but it is not in the ring if the coefficients are complex and not rational.
Q2:
The real answer that I am after is for the general case where $p(x)$ is reducible. For example when $p(x)=x^n-1$, then if $$p(x)=\prod_{i=1}^dp_i(x)$$ for different minimal polynomials for the roots $\{\omega_1,\dots,\omega_d\}$ with $\mathcal{R}$ defined as before, then what is $q(x)$ given $\{q(\omega_1),\dots,q(\omega_d)\}$? is there a closed formula? (it is easy to see that also here $q$ is uniquely defined.)
Thank you very much in advance!