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Why can we consider the prime ideals of $\Bbb Z$ to determine the prime ideals of $\Bbb Z[x]$?

There really isn't any work I can show here. Motivation is that it seems various papers consider a prime ideal of $\Bbb Z$ and then consider cases to single out what this prime ideal of $\Bbb Z$ looks like and then they say these are the prime ideals of $\Bbb Z[x]$

Nick Atwood
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  • Just look there http://math.stackexchange.com/questions/174595/classification-of-prime-ideals-of-mathbbzx – marwalix May 31 '15 at 03:30
  • @marwalix It looks like he lets it be a prime ideal of $\Bbb Z[x]$ and then under intersection it is a prime ideal of only $\Bbb Z$? – Nick Atwood May 31 '15 at 03:32
  • Given a prime ideal $P$ of $\Bbb Z[x]$, most of what you want to know can be discovered by considering $P \cap \Bbb Z$, yes. – David Wheeler May 31 '15 at 03:36
  • So the initial $P$ should be in $\Bbb Z[x]$ okay. Thanks @David, that is what I suspected. – Nick Atwood May 31 '15 at 03:36

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