I am just studying for my exam in commutative algebra and I am trying to compute the Hilbert polynomial of the ring $R=\mathbb{Z}_{(2)}[\sqrt{-3}]$.
To do so, I was trying to calculate $\dim_k(\mathfrak{m}^i/\mathfrak{m}^{i+1})$ where $k$ denotes the residue field of $R$ and $m$ its maximal ideal. I already figured out that the maximal ideal is $(2,1+\sqrt{-3})$ by considering the ring as $\mathbb{Z}_{(2)}[X]/(X^2+3)$ and the residue field $k=\mathbb{Z}/2\mathbb{Z}$. However, I have difficulties determining the dimension of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ as a $k$-vector space. Since $\dim_k(\mathfrak{m}/ \mathfrak{m}^{2})$ is the minimal number of generators for the maximal ideal, it should be 2. But I cannot deduce that fact from the representation of $\dim_k(\mathfrak{m}/ \mathfrak{m}^{2})$ as $(2,1+\sqrt{-3})/(4, 2(1+\sqrt{-3}),(1+\sqrt{-3})^2)$.
Any explanation for that would be very helpful. I think I could generalize the behaviour to higher powers.