Every such quadratic field is some where is a (uniquely defined)
square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and, if , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a
subfield of the field of the
real numbers.
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.
For a nonzero square free integer , the
discriminant of the quadratic field is if is congruent to modulo , and otherwise . For example, if is , then is the field of
Gaussian rationals and the discriminant is . The reason for such a distinction is that the
ring of integers of is generated by in the first case and by in the second case.
The quotient ring is the
finite field with elements: .
splits
is a product of two distinct prime ideals of .
The quotient ring is the product .
is ramified
is the square of a prime ideal of .
The quotient ring contains non-zero
nilpotent elements.
The third case happens if and only if divides the discriminant . The first and second cases occur when the
Kronecker symbol equals and , respectively. For example, if is an odd prime not dividing , then splits if and only if is congruent to a square modulo . 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 modulo , where is the field discriminant.
Then, the ideal class group is generated by the prime ideals whose norm is less than . This can be done by looking at the decomposition of the ideals for prime where [1]page 72 These decompositions can be found using the
Dedekind–Kummer theorem.
Quadratic subfields of cyclotomic fields
The quadratic subfield of the prime cyclotomic field
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 th root of unity, with an odd prime number. The uniqueness is a consequence of
Galois theory, there being a unique subgroup of
index in the Galois group over . As explained at
Gaussian period, the discriminant of the quadratic field is for and for . This can also be predicted from enough
ramification theory. In fact, is the only prime that ramifies in the cyclotomic field, so is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants and in the respective cases.
Other cyclotomic fields
If one takes the other cyclotomic fields, they have Galois groups with extra -torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant can be obtained as a subfield of a cyclotomic field of th roots of unity. 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.
Orders of quadratic number fields of small discriminant
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.