# Proving unique factorization for polynomials

• Jul 1st 2010, 10:23 AM
squelchy451
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!
• Jul 1st 2010, 01:02 PM
TKHunny
Start by stating the theorem.
• Jul 1st 2010, 02:53 PM
squelchy451
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
• Jul 1st 2010, 05:35 PM
chiph588@
In general a ring $\displaystyle R$ is a $\displaystyle \text{UFD}\iff R[x]$ is a $\displaystyle \text{UFD}$.
• Jul 1st 2010, 05:50 PM
squelchy451
The number theory textbook i'm using says nothing about rings...ring R = UFD?
• Jul 3rd 2010, 09:04 PM
squelchy451
bump bump bmup
• Jul 3rd 2010, 09:13 PM
roninpro
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?
• Jul 3rd 2010, 09:44 PM
squelchy451
@roninpro yea I think i get it..io'llt ry that out and see how it goes.