Exhibit an algebraic integer in a quadratic integer ring having a given norm and trace

Find an algebraic integer \alpha \in \mathbb{Q}(\sqrt{D}) having norm 31 and trace 17.

Let \alpha = a+b\sqrt{D} and \overline{\alpha} = a-b\sqrt{D}. Now N(\alpha) = \alpha\overline{\alpha} = a^2 - Db^2 and \mathsf{tr}(\alpha) = \alpha + \overline{\alpha} = 2a. We wish to find a, b, and D such that a+b\sqrt{D} is an algebraic integer, a^2-Db^2 = 31, and 2a = 17. If such an integer exists, then a = \frac{17}{2} is a half-integer- in particular, we must have D \equiv 1 mod 4. Substituting, and letting b = \frac{b_0}{2}, we have Db_0^2 = 165 = 3 \cdot 5 \cdot 11. Thus D = 165 and b_0 = \pm 1, so that b = \pm \frac{1}{2}. Indeed, we can verify that N(\frac{17}{2} + \frac{1}{2}\sqrt{165}) = 31 and \mathsf{tr}(\frac{17}{2} + \frac{1}{2}\sqrt{165}) = 17.

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