I showed that if $(x)$ is prime in $\mathbb{Z}[x]$, then since $\mathbb{Z}[x] /(x)$ is isomorphic with $\mathbb{Z}$ and $\mathbb{Z}$ is not a field, so $(x)$ is not maximal. But when I want to show that $(x)$ is prime in $\mathbb{Z}[x]$ I've got stuck... How could I show? The old proofs doesn't give any isomorphic function! my question is a particular case, not a theorem (just for more info)
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Please don't change your question after answers have been posted. Ask a new question instead if need be – Bill Dubuque Jan 02 '19 at 00:59
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You can find proofs in many prior answers, e.g. here and here – Bill Dubuque Jan 02 '19 at 01:15
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@bill Dubuque edited – Jan 03 '19 at 13:32
