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I have the quotient ring defined by $$\mathbb{Z}[i]/\left(i^{2}+3i-1\right)$$ How would I go about showing that the order is finite and then finding said order?

I have started by noting that $$\mathbb{Z}[i]\approx\mathbb{Z}[x]/\left(x^{2}+1\right)$$ and that I can write the quotient ring as $$\mathbb{Z}[x]/\left(x^{2}+1,x^{2}+3x-1\right)$$

I would assume that the next step would be to find out what this quotient ring is and then determine the order from that? Any help on how to solve this would be much appreciated.

Jonathan.Lidcombe
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  • Well $i^2+3i-1=3i-2$ is a prime in $\mathbb Z[i]$ for a start ... – Mark Bennet Nov 16 '17 at 13:30
  • @MarkBennet So far in Artin, there hasn't been any talk of prime ideals (or none that I've noticed). Is $3x-2$ the residue of $x^{2}+3x-1$ in $\mathbb{Z}[x]/\left(x^{2}+1\right)$? – Jonathan.Lidcombe Nov 16 '17 at 13:43
  • I'm sorry, I fail to see how this question is an 'exact duplicate' of the other question? Could you please elaborate? – Jonathan.Lidcombe Nov 16 '17 at 13:46
  • Well $x^2+3x-1-(x^2+1)=3x-2$ so the two are in the same residue class. And you will find that the techniques referred to in the duplicate question address the issues you have here - see eg the example at the end of srivatsan's answer, which does what you need to do here. – Mark Bennet Nov 16 '17 at 13:59
  • @MarkBennet Ok, thank you for clarifying that. I'm having trouble with quotient rings and didn't see those solutions as helping with my problem. I'll check them out in more detail and see if it helps me! – Jonathan.Lidcombe Nov 16 '17 at 14:07

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