Characterize the prime ideals in algebraic integer rings which are also unique factorization domains

Characterize the prime ideals in those algebraic integer rings which are unique factorization domains.


By Theorem 9.4 in TAN, a ring \mathcal{O} 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, (\pi) is a prime ideal if and only if \pi is an irreducible element. To that end, suppose first that (\pi) is a prime ideal. If \pi = \alpha\beta, then \alpha\beta \in (\pi), and so without loss of generality \alpha \in (\pi). So \alpha = \pi\gamma = \alpha\beta\gamma, and so 1 = \beta\gamma; hence \beta is a unit. So \pi is irreducible. Conversely, if \pi is irreducible then it is also prime. If \alpha\beta \in (\pi), then \pi|\alpha\beta. Without loss of generality, \pi|\alpha, and so \alpha \in (\pi).

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