We assumed that all polynomials of degree can be factored into irreducibles. Since we wrote with , we can factor and into irreducibles, and therefore, can be factored into irreducibles.
The original problem was to show that factorization of polynomials with integer coefficients (into irreducibles) exists and is unique. This is what we are doing. (Please see my earlier post stating the overall strategy.)
For the second part, you should try to prove that if is irreducible and divides , then either or . Then, suppose that a polynomial has two factorizations. Use this division fact to show that the two factorizations differ only by a unit (or invertible number).
link is here) and I think it relates to this problem, however I'm not the best at interpreting it into a little more understandable language. Perhaps it may help?
I don't want to get off track here either. However I've been digging around and this is what other people have said about this problem:
I still wasn't able to make sense of all of this, and I don't want to give really off track here either. Can someone make sense of this?The fact that Z[X] is a ufd is due to gauss. It is NOT a euclidean domain however. the crucial concept is to define the "content" of a polynomial as the gcd of the coefficients, then prove the content of a product is the product of the contents. then one can reduce the proof to the case of Q[X].
Can anyone else also confirm?