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
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“You cannot go into any field or wood, but it will seem as if every stone had been turned, and the bark on every tree ripped up. But, after all, it is much easier to discover than to see when the cover is off. It has been well said that “the attitude of inspection is prone.” Wisdom does not inspect, but behold.”
—Henry David Thoreau (1817–1862)