In algebraic number theory, a quadratic field is an algebraic number field of degree two over
is a (uniquely defined) square-free integer different from
Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms.
The class number problem is particularly important.
For a nonzero square free integer
, the discriminant of the quadratic field
is the field of Gaussian rationals and the discriminant is
The reason for such a distinction is that the ring of integers of
The set of discriminants of quadratic fields is exactly the set of fundamental discriminants (apart from
In line with general theory of splitting of prime ideals in Galois extensions, this may be[1] The third case happens if and only if
The first and second cases occur when the Kronecker symbol
The first two cases are, in a certain sense, equally likely to occur as
runs through the primes—see Chebotarev density theorem.
[2] The law of quadratic reciprocity implies that the splitting behaviour of a prime
in a quadratic field depends only on
Determining the class group of a quadratic field extension can be accomplished using Minkowski's bound and the Kronecker symbol because of the finiteness of the class group.
[1] page 72 These decompositions can be found using the Dedekind–Kummer theorem.
A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive
As explained at Gaussian period, the discriminant of the quadratic field is
is the only prime that ramifies in the cyclotomic field, so
is the only prime that can divide the quadratic field discriminant.
If one takes the other cyclotomic fields, they have Galois groups with extra
can be obtained as a subfield of a cyclotomic field of
This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.
The following table shows some orders of small discriminant of quadratic fields.
The maximal order of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant of the field.
The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the square of the determinant of the matrix that expresses a basis of the non-maximal order over a basis of the maximal order.
All these discriminants may be defined by the formula of Discriminant of an algebraic number field § Definition.
For real quadratic integer rings, the ideal class number, which measures the failure of unique factorization, is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.
Some of these examples are listed in Artin, Algebra (2nd ed.