How to prove that pZ is a maximal ideal for the ring of integers?

 Emeritus Sci Advisor PF Gold P: 16,101 For another description, look at what happens if you don't assume(Assume k is algebraically closed if need be.) Consider the real polynomial x²+1. Being degree 2, the zero set of this polynomial "ought" to consist of two points on the real line, counting multiplicity. But, alas, its factorization only includes a single polynomial; a single point of Spec R[x]! But we 'know' that the zero set is really two points: i and -i. This is one of the intuitions you want for higher degree ideals, I think: a degree n prime ideal corresponds in some sense to n individually indistinguishable points. Going back to your Gaussian integer example, the ideal (7) corresponds to two "geometric" points. One way to manifest them is as the two different residue maps to F49 (or to $\overline{\mathbf{F}_7}$, if you prefer). e.g. the "function" i has the value $\sqrt{-1}\in \mathbf{F}_{49}$ at one point, and $-\sqrt{-1}\in \mathbf{F}_{49}$ at the other point (after choosing a square root of -1 in F49, of course). The "function" 7, of course, vanishes at both points. The ideal (2), however, corresponds to a double point.