Marc LeBrun <mlb@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@aol.com
=Jpr2718@aol.com
=Marc LeBrun <mlb@fxpt.com> 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}. [...]
Thanks for the informative answer. (Today I'm just looking at D=2, but this will help me later...) Henry Baker also suggested "algebraic integer". However I objected on the grounds that x,y are restricted here to be non-negative. I thought "algebraic integer" connoted negative and even half-integer x&y. It also didn't seem right to say "ring" when there's no additive inverses. Am I confused or just being obsessively pedantic about this? Thanks!
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Marc LeBrun