In a quadratic field, the quotient of two algebraic integers with the same norm is a quotient of an algebraic integer by its conjugate

Let K = \mathbb{Q}(\sqrt{D}) be a quadratic field and let \alpha,\beta \in K be algebraic integers such that N(\alpha) = N(\beta). Prove that there exists an algebraic integer \gamma such that \alpha/\beta = \gamma/\gamma^\prime. Give a nontrivial example.


Note that 1 = N(\alpha)/N(\beta) = N(\alpha/\beta). We showed in this previous exercise that every element of K having norm 1 is a quotient of an algebraic integer by its conjugate. That proof was constructive.

We saw here that 3+4i and 5 are elements of \mathbb{Q}(i) which have the same norm (25) but which are not conjugates or associates. Following the constructive proof given previously, we see that \frac{3+4i}{5} = \frac{2+i}{2-i}.

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