Mike Stay asked: Are there any useful definitions of greatest integer function for algebraic integers? Me: It might be better to look for analogues of the "round to nearest integer" function. Then there's an obvious candidate: tile the space with Voronoi cells centered at the integers, and map any point in the cell to its center. There are fast algorithms too - see Chap 21 of SPLAG (Sphere Packings, Lattices and Groups, by JHC and me). Neil Sloane