Modular arithmetic with complex modulus.

Asked Dec 25 '16 at 00:01

Active Dec 25 '16 at 00:01

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If $n,a\in\Bbb Z$ and $0<a<n$ then we know that $u\equiv n\bmod a$ iff $n=at+u$ holds for some $t\in\Bbb Z$.

Suppose $a=b+i$ where $i^2=-1$ what is $n\bmod a$ for cases $n\in\Bbb Z$ and when $n\in\Bbb Z[i]$?

Does it have interpretation of smallest remainder?

asked Dec 25 '16 at 00:01

Turbo

Good question, possible duplicate: http://math.stackexchange.com/questions/1213942/modulo-operations-over-gaussian-integers – Ethan Bolker Dec 25 '16 at 00:04
Search on Gaussian Euclidean algorithm to find many prior posts on this topic $,(\Bbb Z[i],$ is know as the Gaussian integers). – Bill Dubuque Dec 25 '16 at 00:14
@BillDubuque I wouldn't have used $\Bbb Z[i]$ without knowing Gaussian Integers. – Turbo Dec 25 '16 at 00:25
This may help you: http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/Zinotes.pdf – Xam Dec 25 '16 at 01:38

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