Let be a squarefree integer, let , and let be the ring of integers in . Show that the division algorithm does not hold in with respect to the field norm on .
We wish to find such that, whenever , we have .
If mod 4, then . Let and , and suppose there exist and such that and . Then we have . Note that if , then is too large; thus and . In fact, , so that . Comparing coefficients, we have a contradiction mod 2. So no such and exist.
If mod 4, then . Let and . Suppose there exist and such that and . Now . Since , we have . Since mod 2, . Now , a contradiction.