Quadratic number field

A quadratic number field is a degree-two extension of $\mathbb{Q}$, necessarily of the form $\mathbb{Q}(\sqrt{d})$ for a square-free integer $d\ne 0,1$.

A fundamental discriminant is an integer that occurs as the field discriminant of a quadratic number field. For square-free $d$, it is $d$ when $d\equiv 1(\mathrm{mod}4)$ and $4d$ otherwise.