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Math Help - Unique Factorization Domain?

  1. #1
    Super Member Bernhard's Avatar
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    Unique Factorization Domain?

    Let  \mathbb{Q}_\mathbb{Z}[x] denote the set of polynomials with rational coefficients and integer constant terms.

    (a) If p is prime in  \mathbb{Z} , prove that the constant polynomial p is irreducible in  \mathbb{Q}_\mathbb{Z}[x].

    (b) If p and q are positive primes in  \mathbb{Z} , prove that p and q are not associates in  \mathbb{Q}_\mathbb{Z}[x]

    I am unsure of my thinking on these problems.

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    Regarding (a) I think the solution is as follows:

    We need to show the p is irreducible in  \mathbb{Q}_\mathbb{Z}[x]

    That is if p = ab for  p, a, b \in \mathbb{Q}_\mathbb{Z}[x] then at least one of a or b must be a unit

    But then we must have p = 1.p = p.1 since p is a prime in  \mathbb{Z} and hence is prime in  \mathbb{Q}_\mathbb{Z}[x]

    But 1 is a unit in  \mathbb{Q}_\mathbb{Z}[x] (and also in  \mathbb{Z} )

    Thus p is irreducible in  \mathbb{Q}_\mathbb{Z}[x]

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    Could someone please either confirm that my working is correct in (a) or let me know if my reasoning is incorrect or lacking in rigour.

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    Help with the general approach for (b) would be appreciated

    Peter
    Last edited by Bernhard; May 16th 2013 at 01:22 AM.
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