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?
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!
Try this on for size
Polynomial ring - Wikipedia, the free encyclopedia (subheading: Factorization in K[X])
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.
For part 1) The following text is from the Wikipedia article "Irreducible polynomial" -Is there a way to best explain this in general terms without citing all of Abel's irreducibility theorem? (Is there an easier way?) roninpro suggested using strong induction on the degree of p. I'm not entirely on top of how to do that.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?
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.".
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?
Could you continue to show this? I'm following along but this is a weakpoint for me.
Here is what I dug up though:I don't know if thats applicable, but I'm not sure how I can break it down as you had mentioned using the induction hypothesis.if P(k) is true (called induction hypothesis), then P(k+1) is true