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Math Help - Failure of Unique Factorization

  1. #1
    Super Member Bernhard's Avatar
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    Failure of Unique Factorization

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

    Prove that the only two units in  \mathbb{Q}_\mathbb{Z}[x] are 1 and -1.

    Help with this exercise would be appreciated.

    Peter
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  2. #2
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    Re: Failure of Unique Factorization

    Hi,
    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

    The above is exactly what you must prove for your second post. First what are the units in your ring? I don't believe your "proof". Attached is a correct discussion:

    Failure of Unique Factorization-mhfrings3.png
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