Given the quotient ring $\displaystyle R = \frac{\mathbb{Z}_q[x]}{\langle P(x)\rangle}$, for a prime $q$ and an irreducible polynomial $P(x)\in \mathbb{Z} [x]$. my question is there exist a $P,q$ and elements $w,\gamma$ from $\mathbb{Z}_q$ so we have $\forall a\in R$: $a(w) = w a(\gamma)$.
I tried to show it using cyclic extensions (choosing $P$ such that the number field $K$ defined by $P$ is cyclic) and by Kummer-Dedekind theorem we choose an totally unramified $q$ ($e=f=1$ and $g=\deg(P)$) in $K$ but I got stuck at the existence/inexistence of such a $w$