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Raising the span of the monic polynomials in ${\rm Z}[x]$ which contain $x^n$?

I am trying to solve the following problem:

Let ${\rm F}[x]$ be the integral domain of fractions of the polynomial ring ${\rm Z}[x]$
and let ${\rm Z}$ be the integral domain consisting of the integers.
We will denote the zero and non-zero elements of ${\rm F}[x]$ by $\zeta$ and $a/b$, respectively, where $\zeta$ is a fixed element of ${\rm Z}$ and $a, b \in {\rm F}[x] \setminus \{0\}$.
Then we define a map ${\rm q} \colon {\rm F}[x] \to {\rm Z}$ by
$${\rm q}(f(x)):=\frac{f(\zeta)}{\pi(f(x))},$$
where $\pi(f(x))$ is the sum of the coefficients of the monic polynomial $f(x)$.
Let ${\rm F}_n$ be the ring of polynomials in ${\rm F}[x]$ which contain $x^n$, let ${\rm Q}_n:={\rm Z}[x]/\langle x^n-\zeta \rangle$ be the quotient and let ${\rm Q}[x]$ be the integral domain consisting of the quotients of

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