A quadratic field is a specific type of number field that is generated by adjoining a square root of a rational number to the field of rational numbers, \(\mathbb{Q}\). More formally, a quadratic field can be expressed in the form: \[ K = \mathbb{Q}(\sqrt{d}) \] where \(d\) is a square-free integer (an integer not divisible by a perfect square greater than 1).
Articles by others on the same topic
A quadratic number field is a number field of degree of a field extension two, necessarily of the form for a square-free integer . Its two field embeddings send to . If , both images lie in the real numbers, giving a totally real number field; if , they form a pair related by complex conjugation, giving an imaginary quadratic field. The ring of integers of a quadratic field depends on modulo four.[ref]