I'm trying to solve a larger problem about maximal and prime ideal, and knowing if $\mathbb{Z}[x]$ is an integral domain would really help me
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Do you know the definition of an integral domain? – lulu May 20 '19 at 23:57
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$\forall a,b\in \mathbb{Z}[x], ab=0 \implies a=0$ or $b=0$ – Mather Guy May 20 '19 at 23:58
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1Right. Can you multiply two non-zero polynomials together to get $0$? – lulu May 20 '19 at 23:59
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2Ahhhhhhh cheers – Mather Guy May 21 '19 at 00:00
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3As an exercise: prove that, for an integral domain $R$, $R[x]$ is always an integral domain. Useful lemma. – lulu May 21 '19 at 00:00
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There is something slightly stronger that is true due to McCoy that you might find interesting regarding zero-divisors in polynomial rings and the relation to the original ring.
Let $F\in R[X]$ be a polynomial over a commutative ring $R$. If $F$ is a zero-divisor then $rF=0$ for some nonzero $r\in R$. The top answer here gives a sketch of the argument: Zero divisor in $R[x]$
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