In this chapter, we present two modular algorithms for factoring in Q[x] and F[x, y] for a field F. The first one uses factorization modulo a “big” prime and is conceptually easier, and the second one uses factorization modulo a “small” prime and then “lifts” it to a factorization modulo a power of that prime. The latter is computationally faster and comprises our most powerful employment of the prime power modular approach introduced in Chapter 5.

Factoring in ℤ[x] and Q[x]: the basic idea

Our first goal is to understand the difference between “factoring in ℤ[x]” and “factoring a polynomial with integer coefficients in Q[x]”. The basic fact is that the latter corresponds to factoring primitive polynomials in ℤ[x], while the former requires in addition the factoring of an integer, namely the polynomial's content. We rely on the following notions which were introduced in Section 6.2.

Let R be a Unique Factorization Domain (our two main applications are, as usual, R = ℤ and R = F[y] for a field F). The content cont (f) of a polynomial f ∈ R[x] is the greatest common divisor of its coefficients (with the convention that the gcd is positive if R = ℤ and monic if R = F[y]).