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?