[GAP Forum] Group of units of finite gaussian integers

[Bill Allombert]

On Thu, Apr 10, 2014 at 06:44:23PM +0000, sopsku wrote:
> Dear Forum,
> 
> How do I define the group of units of Z_n[i] in GAP? In particular I
> ultimately want to determine isomorphism classes of these groups. I assume
> that once I have properly defined the group of units I can then just use
> StructureDescription, but any help in actually defining the isomorphism
> would be appreciated.  

Hello Ron, I do not know how to do it in GAP, but the general solution to this
question is given in Henri Cohen book "Advanced topic in Computational number
theory", GTM 193, Springer, Section 4.2

In your case, it is possible to write down the the group as a product of cyclic
groups if one knows the factorisation of n.
If n is prime then the answer is:
n=2:       C_2
n=1 mod 4: C_(n-1) x C_(n-1)
n=3 mod 4: C_(n^2-1)

If n is squarefree, you can just multiple the above.

In PARI/GP, idealstar(nfinit(x^2+1),n).cyc will gives you the structure of
the group of units of Z_n[i].

Cheers,
Bill.