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I know that there are 9 elements in R, namely $a+b\sqrt{-5}$ with $a,b\in {0,1,2}$. Now I'm trying to find nontrivial ideals of R, and my book tells me that $(1+\sqrt{-5})=\{0,1+\sqrt{-5},2+2\sqrt{-5}\}$. However, I'm confused as to why $(1+\sqrt{-5})\sqrt{-5}=\sqrt{-5}+-5\equiv 2+\sqrt{-5}$ is not in this ideal.

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    $\begingroup$ $-5\equiv 1 (mod 3)$ $\endgroup$ – Mustafa Feb 11 at 9:35
  • $\begingroup$ I love you my friend $\endgroup$ – davidh Feb 11 at 9:38

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