Exhibit a complete residue system for . Plot the points in the complex plane, coloring each point according to which coset of it is in.
Note that . Now each residue class mod has a representative whose norm is strictly less than 5 by the division algorithm. It is easy to see then that the distinct residue classes have representatives in the set . We claim that all but 5 of these are redundant. Indeed:
- and .
Thus a complete residue system is contained in the set . We claim that this set is itself a residue system. To see this, we need to verify that no two are congruent mod . First, since is not a unit, none of , , , and is congruent to 0 mod . Now has norm 2, and thus cannot be divisible by which has norm 5. Similarly, has norm 4, has norm 2, has norm 2, has norm 2, and has norm 4. Thus these representatives are distinct mod . Hence is a complete residue system mod .
Note that . Likewise, we may add any associate to and remain in the same congruence class. If we color the elements congruent to 0 green, to 1 purple, to red, to blue, and to orange, then a small region around the origin in the complex plane appears as follows.