[GAP Forum] Group of units of finite gaussian integers
sopsku
rrburns at cox.net
Thu Apr 10 21:44:24 BST 2014
sopsku <rrburns at ...> writes:
>
> 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.
>
> Thank you for your help.
>
> Ron
>
After some thought I tried working the polynomial ring isomorphic to Z_n[i].
Below is is what I have come up with. Is there a better way to do this in
gap (aside from my simplistic coding)?
n:=7;
x:=Indeterminate(GF(n),"x");
r:=PolynomialRing(GF(n));
p:=x^2+1;
i:=Ideal(r,[p]);
q:=r/i;
List(last);
g:=Units(q);
ele:=List(g);
ord:=Length(ele);
gorders:=Collected(List(ele,Order));
asg:=AllSmallGroups(ord);;
ordersasg:=List([1..Length(asg)],x->Collected(List(asg[x],Order)));
pos:=Position(List(ordersasg,x->x=gorders),true);
StructureDescription(asg[pos]);
Again thanks for any help given.
Ron
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