Marc LeBrun <mlb(a)fxpt.com> wrote:
> Regarding Z*(SqrtD) = {x + y SqrtD; integer x, y, D>=0}...
>
> 2. What's best to call the whole algebraic structure?
For D squarefree and congruent to 2 or 3 modulo 4, it's the ring of algebraic
integers of the field Q(\sqrt{D}) = {x + y \sqrt{D}; x and y rational}. For
D squarefree and congruent to 1 modulo 4, it's the order of conductor 2 in the
ring of algebraic integers of the field. In this case, the ring of algebraic
integers is {x + y(1 + \sqrt{D})/2; x, y integers}.
If D is not squarefree, then D=(f^2)d for some squarefree d, and, depending
on what d is modulo 4, the algebraic structure is the order of conductor f or
2f in the ring of integers of the field Q(\sqrt{d}).
John Robertson
jpr2718(a)aol.com
