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Math Help - Unique Factorization

  1. #31
    Senior Member roninpro's Avatar
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    Quote Originally Posted by Brimley View Post
    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?
    Yes, I wrote it myself in LaTeX. I'd be happy to post the source if you're interested in it.

    My goal was to make this exposition as self-contained as possible. I only assumed that you knew about some basic concepts: polynomial rings, irreducibility, etc... Because of this, I require the statement of many theorems and must prove each one with some detail.

    The part of this proof that should be most familiar to you is part one: showing that \mathbb{Q}[x] is a unique factorization domain. If not, I strongly recommend that you go back and examine the proof of the Fundamental Theorem of Arithmetic. You will find that this is a direct analogue. The "hard" part is really part two: showing that a polynomial in \mathbb{Z}[x] can be factored if and only if it can be factored in \mathbb{Q}[x].
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  2. #32
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    I really look forward to seeing the 2nd part of this!
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