I am preparing for my algebra exam, and I am stuck with this problem:
Find all maximal and prime ideals in $\mathbb{Z}[x,y]$ containing the ideal $\mathbb{I}=(55,x^2+4,y).$
What is the general method for this type of problem? Or do we need to resort to some ad-hoc method here?
Any help is appreciated.
The maximal/prime ideals containing $I$ correspond precisely to the maximal/prime ideals of the quotient ring $\Bbb{Z}[x,y]/I$. This quotient is finite, and hence it has only finitely many maximal/prime ideals.
Of course a maximal/prime ideal that contains $55$ must contain either $5$ or $11$, but not both. So you could also look at the two quotients by the ideals $I_1=(5,x^2+4,y)$ and $I_2=(11,x^2+4,y)$ separately, where $$\Bbb{Z}[x,y]/I_1\cong\Bbb{F}_5[x]/(x^2+4) \qquad\text{ and }\qquad \Bbb{Z}[x,y]/I_2\cong\Bbb{F}_{11}[x]/(x^2+4).$$ Can you determine the maximal/prime ideals of these two finite rings?