Let $\displaystyle D \in \mathbb{Z}$ be square free. Consider $\displaystyle K = \{ x| ax^2+bx+c = 0, a,b,c \in \mathbb{Z} \}$.

For which conditions on $\displaystyle a,b,c\in \mathbb{Z}$ is $\displaystyle K= \{ u + v\sqrt{D}: u,v \in \mathbb{Q} \}$?