I looked in a large handful of algebraic number theory books, and found only one (see [1, §6.2]) that reports on how to solve the general problem of computing the maximal ideals of over a given prime . In general, this appears to be a surprising problem, in the sense that the algorithms to solve it are much more sophisticated than Theorem 2.1. However, these complicated algorithms all run very quickly in practice.
For simplicity we consider the following slightly easier problem, whose solution contains the key ideas. Let be any order in , i.e., a subring of such that the additive abelian group is finite. Let .
Let be a number field given by an algebraic integer as root of its minimal monic polynomial of degree . We assume that an order has been given by a basis and that that contains .
Given a prime number , the following (sketch of an) algorithm computes the primes lying over , i.e., the maximal ideals of that contain . Each of the following steps can be carried out very efficiently using little more than linear algebra over . The details are in [1, §6.2.5].