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Problem (Irreducibles behave like prime numbers in $Z[x]$)

Suppose that $p(x)\in \mathbb{Z}[x]$ is irreducible over $\mathbb{Z}$.

  1. Prove that if $p(x)\mid a_1(x)a_2(x)$ where $a_1(x),a_2(x)\in \mathbb{Z}[x]$, then we must have either $p(x)\mid a_1(x)$ or $p(x)\mid a_2(x)$.
  2. Use induction to show that if $p(x)\mid a_1(x)a_2(x)\cdots a_n(x)$ where $a_i(x)\in \mathbb{Z}[x]$, then we must have $p(x)\mid a_i(x)$ for some $i$.

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