Quadratic Field

In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q(√d) is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields. If d > 0 the corresponding quadratic field is called a real quadratic field, and for d < 0 an imaginary quadratic field or complex quadratic field, corresponding to whether its archimedean embeddings are real or complex.

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.