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Math Help - Polynomial Rings and UFDs

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    Super Member Bernhard's Avatar
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    Polynomial Rings and UFDs

    Exercise 1, Section 9.3 in Dummit and Foote, Abstract Algebra, reads as follows:

    Let be an integral domain with quotient field and let be monic. Suppose factors nontrivially as a product of monic polynomials in , say , and that . Prove that is not a unique factorization domain. Deduce that is not a unique factorization domain.
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    I found a proof on Project Crazy Project which reads as follows:

    Suppose to the contrary that is a unique factorization domain. By Gauss’ Lemma, there exist such that and . Since , , and are monic, comparing leading terms we see that . Moreover, since is monic and , we have . Similarly, , and thus is a unit. But then , a contradiction. So cannot be a unique factorization domain.

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    Question (1)

    Consider the following statement in the Project Crazy Project Proof:

    "Moreover, since is monic and , we have ."

    My reasoning regarding this statement is as follows:

    Consider  a(x) = x^n + a_{n-1}x^{n-1} + ... ... + a_1x + a_0 ... ... ... (a)

    Then  ra(x) = rx^n + ra_{n-1}x^{n-1} + ... ... + ra_1x + ra_0 ... ... ... (b)

    In equation (b) r is the coefficient of  x^n and coefficients of polynomials in R[x] must belong to R

    Thus  r \in R

    Is this reasoning correct? Can someone please indicate any errors or confirm the correctness.

    ================================================== ======================================


    Question (2)


    The last part of the Project Crazy Project (PCP) reads as follows:

    Moreover, since is monic and , we have . Similarly, , and thus is a unit. But then , a contradiction. So cannot be a unique factorization domain.


    Why does r and s being units allow us to conclude that  a(x) \in R[x] ? (I am assuming that the author of PCP has made an error in writing R in this expression and that he should have written R[x]

    I would be very appreciative of some help.

    Peter
    Last edited by Bernhard; May 12th 2013 at 02:23 AM.
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