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Im assuming $\mathbb{C}$ is the additive group of complex numbers and $\mathbb{Z}[i]$ the gaussian integers with the same operation.

I understand that this is a higher-dimensional use of the basic result that $\mathbb{R}/\mathbb{Z}\simeq S^1$. Geometrically, it makes sense to identify points into a torus, but since we are working with groups i beg to know how does a calculation work formally inside this new structure?

Mathipulator
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  • Welcome to MSE. You should choose your tags carefully. What has this to do with [tag:abstract-algebra]? – José Carlos Santos Apr 27 '23 at 09:55
  • You might be interested by this question. – Jean Marie Apr 27 '23 at 10:03
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    @José Carlos Santos As algebraic structures like the ring of gaussian integers $\mathbb{Z}[i]$ are evoked into the question, I don't find abnormal to use this tag, even if, finaly, the final issue is of a more topological nature. Algebraic-topology tag is indeed more advisable – Jean Marie Apr 27 '23 at 10:07
  • Im asking about how one is able to reduce the real and imaginary parts of a sum of complex numbers in T^2 – Mathipulator Apr 28 '23 at 07:01
  • The OP is asking about the construction of quotient groups, this is part of group theory which is part of abstract algebra. For $H$ a subgroup of a say abelian group $G$ then the elements of $G/H$ are subsets of $G$ of the form $S_a = { a+h,h\in H}$. The addition is $S_a+S_b=S_{a+b}$. Obviously $S_a=S_{a+h}$ for any $h\in H$. – reuns Apr 28 '23 at 09:01
  • Otherwise $\Bbb{C/Z[i]\cong R^2/Z^2\cong (R/Z)^2}$ the isomorphism being to send $a+ib+\Bbb{Z}[i]$ to $(a,b)+\Bbb{Z}^2$ to $(a+\Bbb{Z},b+\Bbb{Z})$. – reuns Apr 28 '23 at 09:02
  • So each a and b in the equivalence classes can be reduced to a lower form by an integer? – Mathipulator Apr 28 '23 at 14:42

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