In B. L. van der Waerden's Algebra it's said that one can considerably simplify usage of Kronecker's method for polynomials over the ring of integers by factoring the given polynomial modulo 2 and possibly modulo 3, so that one gets an idea what degrees the possible factor polynomials might have, and to what residue classes the coefficients modulo 2 and 3 might belong.
I don't have a clue how that information might help. Can someone explain that? I would be really grateful for a little example too, that would help me for good.
Moreover, I might be wrong in understanding what is a polynomial modulo ring element. Is that just polynomial with coefficients modulo that element?
Best Answer
Suppose that $p(x)$, with integer or rational coefficients, factors over the rationals into $p_1(x)p_2(x)...p_k(x)$. Then, modulo $q$, it must factor in at least $k$ factors (not necessarily of the same degree).
So, the idea is that factorizations modulo $q$ gives you a bound from below on the number of irreducible factors over the rationals. In the extreme case that you find that modulo $q$ the polynomial is irreducible of the same degree, then the polynomial is forced to be irreducible over the rationals.