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Math Help - Unique Factorization

  1. #16
    Senior Member roninpro's Avatar
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    We assumed that all polynomials of degree \leq n can be factored into irreducibles. Since we wrote p(x)=q(x)r(x) with 1\leq \deg q, \deg r\leq n, we can factor q and r into irreducibles, and therefore, p can be factored into irreducibles.
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  2. #17
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    Quote Originally Posted by roninpro View Post
    We assumed that all polynomials of degree \leq n can be factored into irreducibles. Since we wrote p(x)=q(x)r(x) with 1\leq \deg q, \deg r\leq n, we can factor q and r into irreducibles, and therefore, p can be factored into irreducibles.
    So how do we follow up with the second part and prove that this factorization is unique? And how do we tie this into the original problem?
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  3. #18
    Senior Member roninpro's Avatar
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    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 p(x) is irreducible and p(x) divides q(x)r(x), then either p(x)|r(x) or p(x)|q(x). 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).
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  4. #19
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    Quote Originally Posted by roninpro View Post
    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 p(x) is irreducible and p(x) divides q(x)r(x), then either p(x)|r(x) or p(x)|q(x). 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).
    I'm not exactly sure how I can prove/show either part of what is noted above. However, I was reading through this document (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?
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  5. #20
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    Quote Originally Posted by Samson View Post
    I'm not exactly sure how I can prove/show either part of what is noted above. However, I was reading through this document (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?
    Any updates on this? It looks as if you both had a good flow going. I'd like to see the end of this solution.
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  6. #21
    Senior Member roninpro's Avatar
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    Are you familiar with the situation for integers? If p is a prime, can you show that if p|qr then either p|q or p|r? This proof works for your polynomials as well.
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  7. #22
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    Quote Originally Posted by roninpro View Post
    Are you familiar with the situation for integers? If p is a prime, can you show that if p|qr then either p|q or p|r? This proof works for your polynomials as well.
    Isn't this more of something we use in the proof rather than proving this statement? To me it seems obvious that if there is a prime integer p that divides a product qr, then p must divide either q or r. That seems pretty straight forward to me.
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  8. #23
    Senior Member roninpro's Avatar
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    What makes you say that it is obvious? If your reason is the Fundamental Theorem of Arithmetic, it is actually circular! You need to prove this divisibility property to prove the Fundamental Theorem.
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  9. #24
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    Quote Originally Posted by roninpro View Post
    What makes you say that it is obvious? If your reason is the Fundamental Theorem of Arithmetic, it is actually circular! You need to prove this divisibility property to prove the Fundamental Theorem.
    Wow, well I totally am confused by this. How am I supposed to show that then ?
    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:

    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].
    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?
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  10. #25
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    Hey Samson and roninpro, are there any update on this? The thread was going well and now it seems dead.
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  11. #26
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    Quote Originally Posted by Brimley View Post
    Hey Samson and roninpro, are there any update on this? The thread was going well and now it seems dead.
    roninpro PM'd me and when he gets the chance he is going to post a proof so the logic from each subproof will flow into each other. I'm definitely excited to see this!
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  12. #27
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    //double post
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  13. #28
    Senior Member roninpro's Avatar
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    I'd like to post an incomplete draft of the proof. Maybe it can get you guys started.

    Please let me know if you see anything that isn't right or is unclear.

    Unique Factorization-unique-factorization-draft.pdf
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  14. #29
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    Quote Originally Posted by roninpro View Post
    I'd like to post an incomplete draft of the proof. Maybe it can get you guys started.

    Please let me know if you see anything that isn't right or is unclear.

    Click image for larger version. 

Name:	Unique Factorization Draft.pdf 
Views:	22 
Size:	57.1 KB 
ID:	19781
    Wow! Did you really write that yourself? I mean, how do you even write it in such styling? It makes sense thus far although it is pretty advanced, but I'm following. Now here is the kicker: does proving this need to be this complicated? Does a simpler method exist? I only ask this because when the OP asked the question I thought who ever answered it would post a 1-2 paragraph reply, I never imagined it this detailed!

    Can anyone else also confirm?
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  15. #30
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    Quote Originally Posted by roninpro View Post
    I'd like to post an incomplete draft of the proof. Maybe it can get you guys started.

    Please let me know if you see anything that isn't right or is unclear.

    Click image for larger version. 

Name:	Unique Factorization Draft.pdf 
Views:	22 
Size:	57.1 KB 
ID:	19781
    Awesome! It looks good so far! Since you posted this a few days ago, have you by chance finished the rest yet? I can't wait to see it all come together.
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