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I need a hint (just a hint please, not a full answer) to proving that $(2, 1+\sqrt{-5})$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$. I'm trying to prove it via definition of a prime ideal and deriving equations but it just gets too complicated and equations are just so long. Anyone might suggest another way?

Thanks

user26857
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rie
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  • Maybe there's a way proving that $\mathbb{Z[\sqrt{-5}]/(2, 1+\sqrt{-5})}$ is an integral domain but im not being able to visualize the quotient ring that easily – rie Apr 09 '16 at 23:42
  • Haven't tried this, but you could try finding the homomorphism with the given ideal as the kernel. Apply the first isomorphism thm and check if the image is a domain. – user217285 Apr 09 '16 at 23:45
  • @Nitin That's nice, I'll give it a try :) – rie Apr 09 '16 at 23:49
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    $\mathbb{Z}[\sqrt{-5}]\simeq \mathbb{Z}[X]/(X^2+5)$, so what is the quotient $\mathbb{Z}[\sqrt{-5}]/(2,1+\sqrt{-5})$? – carmichael561 Apr 09 '16 at 23:54
  • @carmichael561 Wouldn't it be easier to use this isomorphism $\mathbb{Z}[\sqrt{-5}] \simeq \mathbb{Z}[X]/(X^2-2X+6)$? Then im guessing $\mathbb{Z}[\sqrt{-5}]/(2, 1+\sqrt{-5})$ would just be $\mathbb{Z}[X]/((X^2-2X+6)*(X-2))$? – rie Apr 10 '16 at 00:10
  • You could also show that the ideal is maximal (which it happens to be): just add some element not in the ideal, and show that what you get is the whole ring. – Mathmo123 Apr 10 '16 at 00:20

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