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If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?

Chris
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    Assuming you mean the subscript to mean "modulo" the number $m_1+m_2i$, the answer is just the norm: $m_1^2+m_2^2$ – Adam Hughes Jul 22 '14 at 00:37
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    What do you mean by $(\Bbb Z[i])_m$? If it is what Adam is saying, see this. – Pedro Jul 22 '14 at 00:37
  • @PedroTamaroff m stands for the modulus – Chris Jul 22 '14 at 00:38
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    @Chris That doesn't clear things up, really. Well, maybe. Do you mean $\Bbb Z[i]/(m)$? – Pedro Jul 22 '14 at 00:39
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    @PedroTamaroff Are you really asking OP if he means the $m$-adic completion, which is necessarily infinite? You know what he means, and suggesting the correct meaning might help him communicate his question, rather than frustrating him. – Thomas Andrews Jul 22 '14 at 00:44
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    In particular, quite a lot of books use $\mathbb Z_n$ to mean the integers modulo $n$, and likewise for Gaussian integers. To confuse him because he isn't using your preferred notation seems unkind. @PedroTamaroff – Thomas Andrews Jul 22 '14 at 00:47
  • @PedroTamaroff yes thats what I mean – Chris Jul 22 '14 at 00:49
  • You might find this previous discussion useful: http://math.stackexchange.com/questions/23358/quotient-ring-of-gaussian-integers/874223#874223. – vociferous_rutabaga Jul 22 '14 at 00:51
  • @ThomasAndrews No, I wasn't asking about the $m$-adic completion. I had a very good guess of what he meant, but I didn't want to assume it was that until the OP clarified. Notations vary wildly! – Pedro Jul 22 '14 at 01:00
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    Right, but look at the question What could he mean? He came here for help, help him, don't ask cryptic questions. Do you think he knows what you mean by changing $m$ to $(m)$? Just ask, "Do you mean $\mathbb Z[i]$ modulo $m$?" – Thomas Andrews Jul 22 '14 at 01:02

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