[For the entire collection see Zbl 0635.00018.]
The authors give an overview on methods for factorizing polynomials with (algebraic) integer coefficients. If the polynomial coefficients are rational integers this problem can indeed be solved efficiently by factoring the polynomial modulo a suitable (small) prime number p, lifting coprime factors by Hensel’s lemma to a sufficiently large power of p - say \(p^ n\)- and testing whether any of the factors modulo \(p^ n\) yields a factor over the rational integers.
Factoring in algebraic number fields is much more complicated. The authors discuss various methods developed recently which mostly generalize the methods applied over the rational integers. It seems that every proposed algorithm is not very efficient for certain specific polynomials. One major difficulty arises for example, if an integral basis of the field under consideration is not known. Hence, there is still need for a good general factoring algorithm over algebraic number fields. The paper contains an extensive bibliography on that subject which is about complete up to 1986.