Mathematics and Algorithms for Computer Algebra

The motivation for computing in a suitably chosen homomorphic image of the original problem domain is that the computation there is either easier or is possible, whereas in the original domain it was either harder or impossible. For example, any homomorphic image of the infinite ring of integers is a finite ring, and computation in a finite set is usually easier than in an infinite set; as we have already seen, a finite algorithm to find the solution of an equation over a finite set is, at least in principle, to simply test each element of the set, whereas over an infinite set this procedure does not give a finite algorithm. Moreover, there are homomorphic images of the integers that are fields although the integers themselves do not form a field, and finite fields are easy computational domains because division by any nonzero element is possible, but there is no need for frequent gcd computations as there is in a quotient field (field of fractions). Computing in a homomorphic image means computing in a quotient ring, i.e. modulo some ideal in the original ring. Frequently, the original problem ring is the ring of integers Z, and hence the homomorphic image is Zm for some m ∈ Z, which might well be chosen to be prime.