In algebraic number theory, one works in extensions of Z by algebraic integers and studies how the prime from Z factor (possibly into ideals when unique factorization fails) in this larger ring.  We will show how to how to extend Z in such a way that we can control the primes in any Pi_2 way, all while maintaining unique factorization.  As a corollary, we establish the existence of a computable UFD such that the set of primes is Pi_2-complete in every computable presentation.