In
algebraic number theory, a quadratic field is an
algebraic number field of
degree two over
, the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s.
Every such quadratic field is some
where
is a (uniquely defined)
square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.
Quadratic fields have been studied in great depth, initially as part of the theory of
binary quadratic form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results are ...
s. There remain some unsolved problems. The
class number problem is particularly important.
Ring of integers
Discriminant
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 rational
In mathematics, a Gaussian rational number is a complex number of the form ''p'' + ''qi'', where ''p'' and ''q'' are both rational numbers.
The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(''i''), obtained b ...
s 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 set of discriminants of quadratic fields is exactly the set of
fundamental discriminant In mathematics, a fundamental discriminant ''D'' is an integer invariant (mathematics), invariant in the theory of integer, integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ...
s.
Prime factorization into ideals
Any prime number
gives rise to an ideal
in the
ring of integers of a quadratic field
. In line with general theory of
splitting of prime ideals in Galois extensions In mathematics, the interplay between the Galois group ''G'' of a Galois extension ''L'' of a number field ''K'', and the way the prime ideals ''P'' of the ring of integers ''O'K'' factorise as products of prime ideals of ''O'L'', provides one ...
, this may be
;
is inert:
is a prime ideal.
: The quotient ring is the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
elements.
The third case happens if and only if
divides the discriminant
. The first and second cases occur when the
Kronecker symbol
In number theory, the Kronecker symbol, written as \left(\frac an\right) or (a, n), is a generalization of the Jacobi symbol to all integers n. It was introduced by .
Definition
Let n be a non-zero integer, with prime factorization
:n=u \cdot ...
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
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal ...
.
The law of
quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
implies that the splitting behaviour of a prime
in a quadratic field depends only on
modulo
, where
is the field discriminant.
Class group
Determining the class group of a quadratic field extension can be accomplished using
Minkowski's bound
In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number field
In mathematics, an algebraic number field (or simply number field) is an extension ...
and the
Kronecker symbol
In number theory, the Kronecker symbol, written as \left(\frac an\right) or (a, n), is a generalization of the Jacobi symbol to all integers n. It was introduced by .
Definition
Let n be a non-zero integer, with prime factorization
:n=u \cdot ...
because of the finiteness of the
class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
. A quadratic field
has
discriminant
so the Minkowski bound is
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
page 72 These decompositions can be found using the
Dedekind–Kummer theorem
In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure.
Statement for number fields
Let K
be a number field
In mathematics, an algebraic number f ...
.
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
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
generated by a primitive
th root of unity, with
an odd prime number. The uniqueness is a consequence of
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, there being a unique subgroup of
index in the Galois group over
. As explained at
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier tra ...
, 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 In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by for abelian extensions and by for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension L/K of l ...
.
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 .
For real quadratic integer rings, the
ideal class number, which measures the failure of unique factorization, is given i
OEIS A003649 for the imaginary case, they are given i
OEIS A000924
Some of these examples are listed in Artin, ''Algebra'' (2nd ed.), §13.8.
See also
*
Eisenstein–Kronecker number
*
Heegner number
*
Infrastructure (number theory)
*
Quadratic integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
:
with and (usual) integers. When algebra ...
*
Quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducibl ...
*
Stark–Heegner theorem
*
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
*
Quadratically closed field
Notes
References
* Chapter 6.
*
**
* Chapter 3.1.
External links
*
*{{springerEOM, title=Quadratic field, id=Quadratic_field&oldid=25501
Algebraic number theory
Field (mathematics)