Milnor Number for holomorphic map germ Definition: Let $f : (\mathbb{C}^{n}, p) \longrightarrow (\mathbb{C}^{n}, q)$ be a holomorphic map germ. The multiplicity of $f$ at $p$, or Milnor number de $f$ at $p$, noted  $\mu_{p}(f)$, is the dimension of the $\mathbb{C}$-linear space $\mathcal{Q}_{f}$.
Here, we have:
1) $\mathcal{Q}_{f} = \dfrac{\mathcal{O}_{p}}{\mathcal{I}_{f}}$, where : $\mathcal{O}_{p}$ denotes the (local) ring germs of holomorphic functions at $p \in \mathbb{C}^{n}$ and $\mathcal{I}_{f}$ is the ideal in $\mathcal{O}_{p}$ generated by $f_{1}, \cdots , f_{k}$ $(f : (\mathbb{C}^{n}, p) \longrightarrow (\mathbb{C}^{k}, q))$
Consider the germ $f : (\mathbb{C}^{3}, p) \longrightarrow (\mathbb{C}^{3}, 0)$ with $p = (i, 0, 0)$ and for $z = (z_{1}, z_{2}, z_{3})$ the coordinate functions thus defined:
2)$f_{1}(z) = z_{2} + z_{1}z_{3}$,   $f_{2}(z) = - z_{2}z_{3}$,     $f_{3}(z) = z_{1}^{2} + z_{2}^{2} + 1$ 
I'm having trouble calculating a dimension of the $\mathbb{C}$-linear space of definition above. Any help is very welcome.
Thank you very much.

Expand the function as a power series around $(i,0,0)$
$$
f_1 = z_2 +z_1z_3 =z_2 +(z_1-i+i)z_3 = z_2 +iz_3 + (z_1-i)z_3\\
f_2 = -z_2z_3 \\
f_3 = z_1^2+z_2^2+1 = (z_1-i+i)^2+z_2^2+1 = 2i(z_1-i) + (z_1-i)^2+z_2^2
$$
$\mathcal{O}_{(i,0,0)}= \mathbb{C}\{z_1-i, z_2, z_3\}$ is the ring of convergent power series in the variables $z_1-i, z_2, z_3$.  we see that $z_2$ does not belong tho the ideal $(f_1,f_2,f_3)$ then its class in $\mathcal{Q}_f$ is not zero. Now take $\mathcal{Q}_f/(\overline{z_2})$. We have that
$$
\mathcal{Q}_f/(\overline{z_2}) = \left(\frac{\mathbb{C}\{z_1-i, z_2, z_3\}}{(z_2 +iz_3 + (z_1-i)z_3,-z_2z_3, 2i(z_1-i) + (z_1-i)^2+z_2^2)} \right)/(\overline{z_2})= \frac{\mathbb{C}\{z_1-i,z_2 , z_3\}}{(z_2 +iz_3 + (z_1-i)z_3,-z_2z_3, 2i(z_1-i) + (z_1-i)^2+z_2^2,z_2)} = \frac{\mathbb{C}\{z_1-i,  z_3\}}{(iz_3 + (z_1-i)z_3, 2i(z_1-i) + (z_1-i)^2)} = \frac{\mathbb{C}\{z_1-i,  z_3\}}{(z_3, z_1-i)} = \mathbb{C}
$$
Hence $\mathcal{Q}_f = \mathbb{C}\oplus \mathbb{C}\overline{z_2}$.
