Let be considered as an ideal in . Find a generator and a basis for . Draw a diagram to visualize the elements of on the complex plane.
Note that and , so that . Conversely, since , we have . Hence .
We claim that is a basis for over . To see this, note that , so that ; certainly , so these sets are equal. Now suppose . Comparing coefficients, we have and , so that . Thus is free as a generating set for over .
We can visualize this ideal in the complex plane as in the following diagram.