we know that the maximal ideals of ${\mathbb Z}[x]$ are of the form $(p, f(x))$ where $p$ is a prime number and $f(x)$ is a polynomial in ${\mathbb Z}[x]$ which is irreducible modulo $p$.


Is it true that:


the maximal ideals of ${\mathbb Z}[x,y]$ are of the form $(p, f(x,y),g(x,y))$ where $p$ is a prime number and $f(x,y), g(x,y)$ are polynomials in ${\mathbb Z}[x,y]$ which are irreducible modulo $p$.



Maximal ideals of Z[x,y]
Better yet, you can replace $f(x,y)$ with $f(x)$. See the answer to [this question](https://mathoverflow.net/questions/26497/maximal-ideals-in-the-ring-kx1-xn).


**Edited to add:** At Martin Brandenburg's request, I'm expanding this to add the details I thought were too obvious to mention: 


1) A maximal ideal $M$ of ${\mathbb Z}[X,Y]$ is the kernel of a map to a field $k$. 


2) Any field of characteristic zero contains ${\mathbb Q}$ and hence is not finitely generated as a ${\mathbb Z}$-algebra.


3) Therefore the field $k$ has finite characteristic $p$; therefore $M$ contains $p$.


4) Now $M/(p)$ is a maximal ideal in $({\mathbb Z}/p{\mathbb Z})[X,Y]$ and therefore (by the answer to the question linked above) has the form $(\overline{f}(X),\overline{g}(X,Y))$.


5) We can lift $\overline{f}$ and $\overline{g}$ to polynomials $f,g\in M$. 


6) It is easy to check that $p,f,g$ generate $M$.


7) Because ${\overline f}(X,Y)={\overline f}(X,0)$, it follows that $f(X,Y)-f(X,0)$ maps to zero mod $p$. 


8) By 7) and 6), $(p,f(X,0),g(X,Y))=(p,f(X,Y),g(X,Y))=M$, so that $M$ has generators of the advertised form. 


**Edited to add further:** As Yves Cornulier points out in comments, step 2) above is less trivial than both I and Will Sawin made it out to be. The key additional point is that a field $k$ finitely generated over ${\mathbb Z}$ must have finite characteristic because --- by the generalized Nullstellensatz --- the unique closed point in $Spec(k)$ must map to a closed point in $Spec({\mathbb Z})$. 


