=Richard Schroeppel <rcs@CS.Arizona.EDU> There's another well-known concept of "positive" for algebraic numbers. I think it's named "totally positive", and it means the number is positive (in the reals) for all choices of embedding: I.e., all conjugates are positive. 1+sqrt2 fails here because the alternate embedding sqrt2 -> -1.414 gives a negative value (-.414). But 2+sqrt2 is totally positive. As is 2-sqrt2. It seems that totally positives are closed under + and *.
That's an interesting concept. The slant, emphasizing the "rootiness" of a large set of algebraics, is completely different from my recent obsession, which has been trying to isolate and understand the incremental ramifications of minimalist extensions (here just adjoining sqrt2). When you allow negative components you get dense sets that are much like Q. Avoiding them, the resulting elements can be indexed in order...