I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question.
It is clear to me that any root of a polynomial in R allows to factor the polynomial it by X minus the root. But when the root is in one of R's extensions, then X minus the root isn't a polynomial over R anymore, and factoring in R[X] loses its sense. But in some sense, which I do not understand, there seems to always be another polynomial, which IS in R[X], and that cancels out at the same root, by which you CAN factor.
To make this clear, consider for instance that R is ring of integers. Suppose we have a polynomial with 1/2 as a root. Then (X-1/2) isn't in Z[X] so factoring by it isn't suitable. Then, though, I heard a claim that any polynomial in Z[X] that has 1/2 as a root can be factored by (2X-1). But why is that true, cause saying that we can factor by any polynomial that has 1/2 as a root is clearly wrong. So how do we know that 2X-1 is a good choice ? I mean, if we chose multiples of (2X-1) we would reach a point where the coefficients of the cofactor can't be further divided. Can anyone explain ?
Thanks !
Dan