Hello All,

Can someone prove unique factorization for the set of polynomials in with integer coefficients?

I know that the analogous Euclidean algorithm functions for these polynomials using "degree" instead of "norm."

Any help is appreciated!

Printable View

- Sep 9th 2010, 02:14 PMSamsonUnique Factorization
Hello All,

Can someone prove unique factorization for the set of polynomials in with integer coefficients?

I know that the analogous Euclidean algorithm functions for these polynomials using "degree" instead of "norm."

Any help is appreciated! - Sep 17th 2010, 09:19 AMSamson
To me, this problem seems like it would be easy given the wording of the question and knowing that the Euclidean algorithm applies for these polynomials. Is this more complex ? Can anyone offer any help?

- Sep 17th 2010, 09:54 AMundefined
Try this on for size

Polynomial ring - Wikipedia, the free encyclopedia (subheading: Factorization in K[X]) - Sep 17th 2010, 05:53 PMSamson
I looked at it but that didn't seem to get me too far. What was it that I was supposed to notice? I failed to see how it connected for the set of polynomials in x.

- Oct 31st 2010, 10:25 PMBrimley
- Nov 2nd 2010, 10:17 PMSamson
- Nov 8th 2010, 02:49 PMundefined
Well it turns out the link I gave is not directly applicable since is not a field. But is a unique factorization domain (UFD), and it can be proven that the ring of polynomials over a UFD is also a UFD. Apologies for not noticing this.

I am looking into the best way to present a proof and/or give a reference. It should be noted that I am not an expert in this but am also learning, as time permits. I try to make sure everything I post is well-reasoned and accurate, but of course, it doesn't always work out quite like that. :)

I hope to post again before too many hours pass with something useful; I am busy at the moment. - Nov 8th 2010, 03:28 PMBrimley
- Nov 8th 2010, 04:32 PMroninpro
This is a two-step procedure.

1) Show that any polynomial can be factored into irreducibles.

2) Show that the factorization is unique.

Try part one first. The trick is to use strong induction on the degree of . - Nov 8th 2010, 05:05 PMBrimley
For part 1) The following text is from the Wikipedia article "Irreducible polynomial" -

Quote:

Every polynomial p(x) in F[x] can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by constants from F (because the ring of polynomials over a field is a unique factorization domain).

Any polynomial over F must share either no roots or all roots with any given irreducible polynomial; this is Abel's irreducibility theorem.

For part 2) I found this link here: Unique Factorization of Polynomials

However, it shows this only for a specific example. Is there a more general version of this that is applicable to the OP's case? - Nov 8th 2010, 05:11 PMSamson
Hey roninpro, I am not sure on how to use the strong induction that you had mentioned, although I know what it is. According to Wikipedia [link here], "Another generalization, called complete induction (or strong induction or course of values induction), says that in the second step we may assume not only that the statement holds for n = m but also that it is true for all n less than or equal to m.".

- Nov 8th 2010, 05:12 PMroninpro
For part one, you can check a few base cases for degree 0 and 1. Suppose that every polynomial of degree can be factored into irreducibles. Let be a polynomial of degree . If it is irreducible, then we are done. On the other hand, suppose is reducible. Then, we can find polynomials of positive degree such that . What can be said at this point?

- Nov 8th 2010, 05:15 PMSamson
- Nov 8th 2010, 05:18 PMroninpro
Yes. You use the induction hypothesis on and .

- Nov 8th 2010, 05:22 PMSamson
Could you continue to show this? I'm following along but this is a weakpoint for me.

Here is what I dug up though:Quote:

if P(k) is true (called induction hypothesis), then P(k+1) is true