For each of the fields and , do the following:
- Describe the integers.
- Give an integral basis.
- Describe all the other integral bases.
- Compute the discriminant of the field.
Note that , since and . Since mod 4, the integers in have the form where , is an integral basis (where ), and the discriminant is -7.
Suppose is another integral basis; then we have , where is an invertible matrix over . That is, . Letting , we have and where .
Similarly, since and . Since mod 4, the integers in have the form with , is an integral basis, and the discriminant is 12. If is another integral basis, then there exists an invertible matrix such that . As before, and where .