Let be a quadratic field and let be algebraic integers such that . Prove that there exists an algebraic integer such that . Give a nontrivial example.
Note that . We showed in this previous exercise that every element of having norm 1 is a quotient of an algebraic integer by its conjugate. That proof was constructive.
We saw here that and are elements of which have the same norm (25) but which are not conjugates or associates. Following the constructive proof given previously, we see that .