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Among quadratic integer rings, $\mathbb{Z}[i]$ and $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ have their special names, namely Gaussian integers and Eisenstein integers respectively. I guess this is named so because these rings are particularly more important than other quadratic rings. Moreover, one can completely characterize prime elements of these rings.

I asked my professor (her major is number theory) and she said one reason why these rings are important is that $\mathbb{Z}$,Gaussian and Eisenstein rings are related to elliptic curves. However, it is not very clear to me how.

Why are Gaussian & Eisenstein integers are important? And what is a use of their prime elements?

Rubertos
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  • Gaussian and Eisentein rings are closely related to say cuadratic, cubic and bicuadratic reciprocity, for example. – Pedro Mar 04 '15 at 00:26
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    @PedroTamaroff Why these two rings are particularly important over other quadratic integer rings? I still don't get it.. – Rubertos Mar 04 '15 at 00:27
  • Using congruences in the ring of integers of $\Bbb Z[i]$, one can recover the law of quadratic reciprocity. Using similar ideas in $\Bbb Z[\omega]$, one can recover the law of cubic reciprocity. – Pedro Mar 04 '15 at 00:29
  • The points themselves form nice, symmetric lattices (square and equilateral triangular, to be specific). I don't know enough about quadratic integer rings to be at all specific, but generally lattices are nice to work with, the more symmetric the better. – pjs36 Mar 04 '15 at 00:52
  • Gaussian rings are the gateway to quadratic fields of class number 1, and Eisenstein rings are the gateway to cyclotomic fields. They probably get special names because they were the first to be investigated by the respective mathematicians. – dezakin Mar 04 '15 at 00:56
  • Here's an answer by Bill Dubuque about the usefulness of Gaussian integers: http://math.stackexchange.com/questions/199676/what-are-imaginary-numbers/199771#199771 – Hans Lundmark Mar 04 '15 at 08:40

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