7 Oct
2003
7 Oct
'03
6:21 p.m.
Are there any useful definitions of greatest integer function for algebraic integers? For the Gaussian integers, it's pretty straightforward because it uses the Euclidean norm--find the nearest integer to a complex x with norm less than magnitude(x). It's unique except for some points on diagonals where there are two. What about Z[sqrt(-3)]? I guess one could talk about integers with norm less than magnitude(x), but the magnitude of x is using a Euclidean norm while Z[sqrt(3)] uses a hyperbolic norm, N(a+b sqrt(-3)) = a^2-3b^2. (I think I got that right.) When the greatest integer isn't unique, can one show there are only a finite number of possibilities? -- Mike Stay staym@clear.net.nz http://www.xaim.com/staym