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Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be proved simply using localizations, but can't quite remember how to do it! I definitely don't want a link to a long involved argument about polynomials, I can find quite enough of those!

Many thanks in advance.

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    @Edward: IMO the most difficult part of the argument is that if $I$ is a non-zero prime ideal in $\mathbb{Q}[x]$, then it contains a generator $g$ such that $(g)$ is a prime ideal of $\mathbb{Z}[x]$. This requires some non-trivial facts about polynomial rings: e.g. the fact that $\mathbb{Z}[x]$ is a unique factorization domain, and what the primes elements are.2012-10-25
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    @Hurkyl: It is Gauss' Lemma.2012-10-26
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    There's a wonderful picture by Mumford on this: http://www.neverendingbooks.org/index.php/mumfords-treasure-map.html2012-10-27
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    Related: https://math.stackexchange.com/questions/199990/prime-ideals-in-mathbbzx2017-01-07

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