Let $\mathbb{K}$ be a field. I need to calculate all the maximal ideals that contain $\mathfrak{a}=(X-1,Y-1,T^6+Z^6+Z^5,X+Z)=(X-1,Y-1,T^6+X^6-X^5,X+Z)=(X-1,Y-1,T^6,X+Z)$ in $\mathbb{K}[X,Y,Z,T]$.
I have tried doing the following:
Let $\mathfrak{b}$ be a maximal ideal which contains $\mathfrak{a}$. As a result, we have that $X-1,Y-1,T^6,X+Z \in \mathfrak{b}$.
Because of this, I have defined $\mathfrak{b}$ in the following way: $$\mathfrak{b}=(X-1,Y-1,T^6,X+Z,f_1(X,Y,Z,T),f_2(X,Y,Z,T),...)$$ where $f_i(X,Y,Z,T) \in \mathbb{K}[X,Y,Z,T]$, $\forall i \geq 1$.
Apart from that, we have these congruences: $$ X \equiv 1 (mod(X-1))$$ $$ Y \equiv 1 (mod(Y-1))$$ $$ Z \equiv -X \equiv -1 (mod(X+Z))$$ So $f_i(X,Y,Z,T)\equiv f_i(1,1,-1,T)=g_i(T) \in \mathbb{K}[T]$, $ \forall i \geq 1$. Therefore, $$\mathfrak{b}=(X-1,Y-1,T^6,Z+1,g_1(T),g_2(T),...) $$ 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.