* Keith F. Lynch <kfl@KeithLynch.net> [May 14. 2016 07:37]:
Dan Asimov <dasimov@earthlink.net> wrote:
P.S. Now I'd like to know the same thing for the Gaussian integers Z[i] and the Eisenstein integers Z[w] where w = exp(2pi*i/3). These apparently have some complicated factor rings.
I'm not sure what you mean by modular arithmetic on Gaussian integers.
[...]
Purely computationally, I find the following the most natural approach as it is the same for finite fields. Consider complex numbers as polynomials modulo the polynomial x^2 + 1 (Gaussian) or x^2 + x + 1 (Eisenstein). For Gaussian integers think i=x (a + i*b |--> a + b*x), for Eisenstein integers think w=x (a + w*b |--> a + b*x). An attempt to write up this idea is Section 39.12 "Complex modulus: the field GF(p^2)" of my (fxt)book. What is not in the fxtbook: For computations mod a given Gaussian integer A + i*B use the minimal polynomial P = x^2 - 2*A*x + (A^2 + B^2) as the modulus (is this correct?). For a given Eisenstein integer, OOPS, someone please fill me in (long ago that I looked at this, plus I claim lack of coffee, there also was a traffic jam, and an earthquake, somewhere)! Best regards, jj