Let $p$ be a prime and $g(x)$ irreducible $mod \ p$ over $\mathbb{Z}$. And let $d\ne 1 \ mod \ 4$ be a squarefree integer with $\omega:=\sqrt{d}$.
We know that $(p,g(\omega))$ is a prime ideal over $R:=\mathbb{Z[\omega]}$ and we want to figure out if $(p,g(\omega))$ is also a maximal ideal, which I tried to proof the following way:
\begin{equation} R/((p,g(\omega))=\mathbb{Z[\omega]}/(p,g(\omega)) \cong \mathbb{Z_p[\omega]}/(g(\omega)) \cong \mathbb{Z_p}^{deg(g)} \end{equation}
Hence $R/((p,g(\omega))$ is a field, which then implies that $((p,g(\omega))$ is a maximal ideal.
Is the last step of the isomorphisms a legit way to deal with $\mathbb{Z_p[\omega]}/(g(\omega))$ or does this idea only apply to $\mathbb{Z_p[x]}/(g(x)) \cong \mathbb{Z_p}^{deg(g)}$? And maybe another question: Is a polynomial ring just a special case of a ring extension?