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Math Help - Coprime Polynomials

  1. #1
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    Coprime Polynomials

    If a(x) and b(x) are coprime in \mathbb{Z}_n[x] and a(x) \mid f(x), b(x) \mid f(x) in \mathbb{Z}_n[x], then is it true that a(x)b(x) \mid f(x) in \mathbb{Z}_n[x]?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    You have ac+bd=1 for some c,d. Write f=aa'=bb', which you may, by assumption. Then we have f=bb'ac+aa'bd = ab(b'c+a'd), so ab|f. You can see that this is, in fact, true in any commutative ring.
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