Show that is an ideal in the ring of algebraic integers in . Find a generator and a basis for this ideal.
First, since mod 4, the elements of are certainly algebraic integers. (Integers in have the form with .) We claim that . To see this, note that if , then . Conversely, if , then . So , and more generally, is an ideal in the ring of integers in .
Now we claim that is a basis of . It is certainly a generating set over . Suppose now that . If , then , so that is rational, a contradiction. So , and we have , so . Thus is -linearly independent, and hence a basis for .