Let $-D\equiv1\mod4$ and $\alpha=\frac{1+\sqrt{-D}}{2}$. Then $\mathbb{Z}[\alpha]$ is the ring of integers of $\mathbb{Q}(\sqrt{-D})$. A general element $\gamma=a+b\alpha$ (for $a,b\in\mathbb{Z}$) of $\mathbb{Z}[\alpha]$ has norm $\mathcal{N}\gamma=a^2+ab+\frac{D+1}{4}b^2$. This quadratic form corresponds to the matrix $$ A= \begin{bmatrix} 1 &\frac{1}{2} \\ \frac{1}{2} &\frac{D+1}{4}\end{bmatrix} $$ which has eigenvalues $\lambda_{\pm}=\frac{(D+5)\pm\sqrt{D^2-6D+25}}{8}$, which are both real. Here are two questions,
- What is the relation between the two sets of values $\{\lambda_+x^2+\lambda_-y^2 | x, y\in\mathbb{Z}\}$ and $\{a^2+ab+\frac{D+1}{4}b^2 | a,b\in\mathbb{Z}\}$?
- What is the relationship of $\lambda_\pm$ with the field $\mathbb{Q}(\sqrt{-D})$?