I need to find all the maximal ideals that contain $\mathfrak{a}=(X+2Z+1,Y-Z,Z^2+Z+1)$ in $\mathbb{C}[X,Y,Z]$.
I have tried doing the following:
Let $\mathfrak{b}$ be a maximal ideal which contains $\mathfrak{a}$. Therefore, we have that $X+2Z+1,Y-Z,Z^2+Z+1 \in \mathfrak{b}$.
Because of this, I have defined $\mathfrak{b}$ in the following way: $$\mathfrak{b}=(X+2Z+1,Y-Z,Z^2+Z+1,f_1(X,Y,Z),f_2(X,Y,Z),...)$$ where $f_i(X,Y,Z) \in \mathbb{C}[X,Y,Z]$, $\forall i \geq 1$.
Apart from that, we have these congruences: $$ X \equiv -2Z-1 \bmod(X+2Z+1), Y \equiv Z \bmod(Y-Z) $$ So $f_i(X,Y,Z)\equiv f_i(-2Z-1,Z,Z)=g_i(Z) \in \mathbb{C}[Z]$, $ \forall i \geq 1$. Therefore, $$\mathfrak{b}=(X+2Z+1,Y-Z,Z^2+Z+1,g_1(Z),g_2(Z),...) $$ But I don't know which polinomials I should choose so that $\mathfrak{b}$ is maximal...
I would be really thankful if someone could help me.
As leoli says in the comments, this is a problem we should solve geometrically. The maximal ideals containing $\mathfrak{a}$ are precisely the ideals $(X-x_0, Y-y_0, Z-z_0)$ where $(x_0, y_0, z_0)$ is a point of the variety $$V = \{(X,Y,Z) \in \mathbb{C}^3 : X+2Z+1 = Y-Z = Z^2 + Z + 1 = 0\}.$$ So we just have to find this set! I'll leave this up to you -- to check your work, $V$ has exactly two elements.
If you haven't yet proved this general fact about maximal ideals of $\mathbb{C}[X,Y,Z]$, the key fact is that every maximal ideal of $\mathbb{C}[X,Y,Z]$ is of the form $(X-x_0, Y-y_0, Z-z_0)$ for some $(x_0, y_0, z_0) \in \mathbb{C}$. This requires a bit of Nullstellensatz: you need to know that the variety defined by a proper ideal $I$ is nonempty. Then take a point $(x_0,y_0,z_0)$ in the variety defined by a maximal ideal $\mathfrak{m}$, and show that $X-x_0 \in \mathfrak{m}$...