TITLE: Are non-zero prime ideals maximal in the ring of all algebraic integers?
QUESTION [6 upvotes]: In the ring of all algebraic integers (i.e. the integral closure of the integers in the complex numbers), can we say that non-zero prime ideals are maximal? 

It's very hard to find a counter example, it's difficult to find a prime ideal to begin with.

REPLY [5 votes]: That is (almost) correct. Actually by the going-up and going-down theorems,  for any integrally closed domain $A$ and any extension $L$ of its field of fractions, the integral closure $A'$ of $A$ in $L$ satisfies 
$$\dim A=\dim A'.$$
($\dim$ denotes the Krull dimension). 
In the present case you apply this result to $\mathbf Z$ which has dimension $1$: any ring of algebraic integers also has dimension $1$, i.e. any non-zero prime ideal is maximal.