# Thread: Proving unique factorization for polynomials

1. ## Proving unique factorization for polynomials

Hi!

I need to prove that unique factorization works for polynomials. I think it might have something to do with the power of the polynomial and the number of roots...the hint in the problem suggested using the Euclidean Algorithm.

Thanks!

2. Start by stating the theorem.

3. you mean state the euclidean algorithm? It's a way of finding the GCD

If you have a and b, the process goes
a = (q1)b + (r1)
b = (q2)(r1) + (r2)
r1 = (q3)(r2) + r3

and keeps going on until the remainder becomes zero

4. In general a ring $\displaystyle R$ is a $\displaystyle \text{UFD}\iff R[x]$ is a $\displaystyle \text{UFD}$.

5. The number theory textbook i'm using says nothing about rings...ring R = UFD?

6. bump bump bmup

7. There are two steps that you need to take. First, you need to show that you can factor every polynomial into a product of primes / irreducibles. Then, you need to show that the factorization is unique.

Can you see how to prove either part?

8. @roninpro yea I think i get it..io'llt ry that out and see how it goes.