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I would like to know whether a polynomial in $\mathbb Z[x]$ is a prime element if and only if it is irreducible.

Since $\mathbb Z[x]$ is an integral domain, a prime element in $\mathbb Z[x]$ is always irreducible. It remains to prove that each irreducible element is a prime. Let $p$ be an irreducible in $\mathbb Z[x]$ and assume $p|ab$. It is clear that $p$ is nonzero and non-unit. Then we have $ab=pc$ for some $c \in \mathbb Z[x]$.

How can I proceed?

user26857
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IDontKnowMath
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    See https://en.wikipedia.org/wiki/Gauss%27s_lemma_(polynomial). – lhf Mar 19 '16 at 11:48
  • Prime = irreducible in any UFD (in fact in any GCD domain); see http://math.stackexchange.com/questions/1595010/prove-that-in-an-integral-domain-if-every-two-elements-have-a-gcd-every-irredu. Do you know that $\mathbb Z[x]$ is a UFD? – user26857 Mar 19 '16 at 12:49
  • @user26857 My prof only proof to us that k[x] is a UFD for any field k. So I don't know... – IDontKnowMath Mar 19 '16 at 13:01

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