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I want to ask somethink. Is it possible to a number, prime in Z[i] but not prime in Z? If yes what's the example? If no, how to prove it?

Dicky Ardiyantoro
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    Unless the number is not in $\Bbb{Z}$, it's not possible: a factorisation in $\mathbf{Z}$ is also a factorisation in $\Bbb{Z}[i]$. – Rob Arthan Mar 27 '21 at 12:47
  • Well, $1+i$ is a Gaussian prime, for example. – lulu Mar 27 '21 at 12:47
  • @lulu yes I know 1+i is Gaussian prime but not prime in Z since it's not element of Z. But is there any element of Z which prime in Z[i] but not prime in Z? – Dicky Ardiyantoro Mar 27 '21 at 12:52
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    Well, no. If $n=ab$ is a composite integer then $n=ab$ is a composite Gaussian integer. – lulu Mar 27 '21 at 12:55
  • Hello @Rob Arthan , I still don't get it why it's not possible, a factorization in Z is also a factorization in Z[i]. Can you explain it more specific or with example? Thank you. – Dicky Ardiyantoro Mar 27 '21 at 12:56
  • Worth remarking, the only integers which are units in $\mathbb Z[i]$ are $\pm 1$. Hence teh factroing $n=a\times b$ with $a,b>1$, is again a factoring into non-unit Gaussian integers. – lulu Mar 27 '21 at 12:57
  • If $n$ is composite in $\Bbb{Z}$, so $n = xy$, where $x, y \in \Bbb{Z}$ and $x$ and $y$ are not units in $\Bbb{Z}$, then $n$ is composite in $\Bbb{Z}[i]$ too because $x$ and $y$ are not units in $\Bbb{Z}[i]$ either (as you can see by considering norms). – Rob Arthan Mar 27 '21 at 13:05
  • Thank you guys, I understand all of your explanation. – Dicky Ardiyantoro Mar 27 '21 at 15:20

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