=Dan Hoey
=Marc LeBrun Regarding Z*(SqrtD) = {x + y SqrtD; integer x, y, D>=0}... What's best to call the whole algebraic structure?
I'd say Z*(SqrtD) = {x + y SqrtD; integer x, y>=0}, for integer D>=0, just to keep the scope straight.
Good point.
I take it "Z*" is the nonnegative integers.
Yes. I've never much liked that notation either. Is there a better one? I've also considered using Z0, N0 and even U (for unsigned) but Z* seems pretty prevalent, despite its opacity.
=Henry Baker "algebraic natural numbers" or "natural algebraic numbers" ??
Cute name, but I'd rather use that for the positive elements of Z(SqrtD) (or the nonnegative ones, in France and set theory).
The "non-negative algebraic integers" might be construed to include the undesired Sqrt2-1, which is positive but has a negative component.
Then the elements of Z(SqrtD) with both components positive could be called "supernatural", and those with at least one component positive could be called "subnatural".
Isn't "natural" supposed to connote a strictly positive integer? I would think "natural algebraic numbers" would refer to only those numbers with all components positive integers. Assuming that, I guess it'd make (some kind of) sense to call the superset got by adjoining the ("unnatural"?<;-) elements with zero components the "supernatural algebraic numbers". So for the particular case that started this all, I should call them the "supernatural quadratic numbers of discriminant 2"? I'm still unsure whether it's acceptable to call their arithmetic a (commutative)(semi)ring, or...what?