Since $\dim R[X] =\dim R+1$ for any Noetherian ring $R$, the ring $\mathbb{Z}[X_1,X_2]$ must have dimension 3. But how can this be proved 'from first principles', i.e. without using any big theorems from commutative algebra?
Clearly the dimension is at least 3, e.g. using the chain $\{0\},(2),(2,X_1),(2,X_1,X_2)$. To prove it's at most 3, I tried assuming the existence of a length-4 chain and then quotienting by the bottom element to get a chain in $\mathbb{Z}[X_1,X_2]/(f)$ for $f$ irreducible, but couldn't get anywhere from there. Can we easily prove something about the structure of the prime ideals in $\mathbb{Z}[X_1,X_2]$?
Many thanks in advance for any help with this!