Algorithms for Algebraic Number Theory

I think the best overall reference for algorithms for doing basic algebraic number theory computations is [Coh93].

Our main long-term algorithmic goals for this book (which we won't succeed at reaching) are to understand good algorithms for solving the following problems in particular cases:

As we will see, somewhat surprisingly it turns out that algorithmically by far the most time-consuming step in computing the ring of integers $ \O _K$ seems to be to factor the discriminant of a polynomial whose root generates the field $ K$. The algorithm(s) for computing $ \O _K$ are quite complicated to describe, but the first step is to factor this discriminant, and it takes much longer in practice than all the other complicated steps.

William Stein 2004-05-06