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Thread: Coprime Polynomials

  1. #1
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    Coprime Polynomials

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