Word Count : 177

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Assume for contradiction that the ideal $(x,5)\subset \mathbb{Z}[x]$ is principal, say $(x,5)=(f)$ for some $f\in\mathbb{Z}[x]$. Then $x\in(f)$ and $5\in(f)$, so $f\mid x$ and $f\mid 5$ in $\mathbb{Z}[x]$. If $f$ is a nonconstant polynomial, then from $f\mid 5$ we would have a factorization $5=f\cdot g$ ______ _____ ___ ____ _____ ____ ____ _____ ____ ________ _____ ______.
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