# 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 dQ(√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 FieldDiscriminant, Prime Factorization Into Ideals

### Other articles related to "quadratic field, fields, quadratic fields, field, quadratic":

Quadratic Field - Quadratic Subfields of Cyclotomic Fields - Other Cyclotomic Fields
... If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields ... In general a quadratic field of field discriminant D can be obtained as a subfield of a cyclotomic field of D-th roots of unity ... This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant ...
Dirichlet's Unit Theorem
... on the idea that there will be as many ways to embed K in the complex number field as the degree n = these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that n ... of α that are real, 2r2 the number that are complex write the tensor product of fields K ⊗QR as a product of fields, there being r1 copies of R and r2 copies of C ... As an example, if K is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field ...
Quaternion Algebras Over The Rational Numbers
... the rational numbers have an arithmetic theory similar to, but more complicated than, that of quadratic extensions of ... A quaternion algebra over the rationals which splits at is analogous to a real quadratic field and one which is non-split at is analogous to an imaginary ... The analogy comes from a quadratic field having real embeddings when the minimal polynomial for a generator splits over the reals and having non-real embeddings otherwise ...

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