Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field of degree two over ${\displaystyle \mathbf {Q} }$, the rational numbers.

Every such quadratic field is some ${\displaystyle \mathbf {Q} ({\sqrt {d}})}$ where ${\displaystyle d}$ is a (uniquely defined) square-free integer different from ${\displaystyle 0}$ and ${\displaystyle 1}$. If ${\displaystyle d>0}$, the corresponding quadratic field is called a real quadratic field, and, if ${\displaystyle d<0}$, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.

Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.