Characterize the prime ideals in those algebraic integer rings which are unique factorization domains.
By Theorem 9.4 in TAN, a ring of algebraic integers from some field is a unique factorization domain if and only if every ideal is principal.
We claim that in a principal ideal domain, is a prime ideal if and only if is an irreducible element. To that end, suppose first that is a prime ideal. If , then , and so without loss of generality . So , and so ; hence is a unit. So is irreducible. Conversely, if is irreducible then it is also prime. If , then . Without loss of generality, , and so .