Let and . Let and be ideals in . For , find a basis for and compute the discriminant and norm of . Compute .
We claim that is a basis for over . Indeed it is clear that , and if , then . Then the discriminant of is and (using Theorem 9.10 in TAN) .
Let be arbitrary, and say and mod 2, where . Certainly mod , so that . Since , we know that these cosets are distinct.
We claim now that is a basis for over . Indeed, note that , and if , then . Moreover, if , then , so that , and thus . Then the discriminant of is and the norm of is .
Note that . If is arbitrary, then mod . If mod 5, with , then mod . Thus . Since , these cosets are distinct.