Let $\mathcal{O}=\mathbb{Z}[\omega]$ be the ring of algebraic integers in $\mathbb{Q}(\omega)$. It can be shown that $\mathcal{O}$ has a maximal ideal $\mathfrak{m}$ generated by $1-\omega$ (see my previous questions).
Let $\mathcal{O}_{\mathfrak{m}}$ denote $\mathcal{O}$ localised at $\mathfrak{m}$.
Why does the residue field $\mathcal{O}_m/(1-\omega)\mathcal{O}_{\mathfrak{m}}$ have characteristic 3?