Editing Quadratic field (section)

{{short description|Field (mathematics) generated by the square root of an integer}}
In [[algebraic number theory]], a '''quadratic field''' is an [[algebraic number field]] of [[Degree of a field extension|degree]] two over <math>\mathbf{Q}</math>, the [[rational number]]s.

Every such quadratic field is some <math>\mathbf{Q}(\sqrt{d})</math> where <math>d</math> is a (uniquely defined) [[square-free integer]] different from <math>0</math> and <math>1</math>. If <math>d>0</math>, the corresponding quadratic field is called a '''real quadratic field''', and, if <math>d<0</math>, it is called an '''imaginary quadratic field''' or a '''complex quadratic field''', corresponding to whether or not it is a [[Field extension|subfield]] of the field of the [[real number]]s.

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