[GAP Forum] Re: Eisenstein integers

[Stefan Kohl]

Dear Forum,

Barry Monson wrote:

> Has anyone developed routines for simple
> computations in the ring of Eisenstein integers,
> i.e.   in   Z + Z*E(3)  ?  I particularly want the
> Gcd function.
>         Or is the ring built into Gap? I
> couldn't make Gap recognize it as a Euclidean domain.

The ring of Eisenstein integers as such is currently not implemented
in the GAP library, but there is support for cyclotomic numbers in
general, thus in particular for the elements of this ring.

Hence it is neither difficult nor much work to implement the ring of
Eisenstein integers as a GAP domain and to install methods for
computing Gcd's etc.

You can easily see how to do this by having a look at the implementation
of the ring of Gaussian integers in the library (see lib/gaussian.gi).

If you have any further questions, please don't hesitate to ask --
as such a discussion might get rather technical, please use the
address support at gap-system.org instead of the whole forum
for this purpose.

Hope this helps,

    Stefan Kohl