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.
Read more about Quadratic Field: Discriminant, Prime Factorization Into Ideals
Famous quotes containing the word field:
“The woman ... turned her melancholy tone into a scolding one. She was not very young, and the wrinkles in her face were filled with drops of water which had fallen from her eyes, which, with the yellowness of her complexion, made a figure not unlike a field in the decline of the year, when the harvest is gathered in and a smart shower of rain has filled the furrows with water. Her voice was so shrill that they all jumped into the coach as fast as they could and drove from the door.”
—Sarah Fielding (1710–1768)