## 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 $\frac{3+4i}{5} = \frac{2+i}{2-i}$.