"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the factors are written."

"Suppose that f = f

_{1}... f

_{r}= g

_{1}... g

_{s}where f is a polynomial over K and f

_{1}... f

_{r}, g

_{1}... g

_{s}are irreducible polynomials over K. If all the f

_{i}are constant then ... so are all the g

_{j}are constant."

So far so good.

"Otherwise, we may assume that no f

_{i}is constant by dividing out all the constant terms."

How can this assumption be made? What if some of the f

_{i}are constant, and some are not constant?

There is more unclear text here. Does anyone have a link to better explanation of this?