Quadratic field
Encyclopedia
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 integer
s 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 form
s. There remain some unsolved problems. The class number problem is particularly important.
of the quadratic field Q(√d) is d if d is congruent to 1 modulo 4, and otherwise 4d. For example, when d is −1 so that K is the field of so-called Gaussian rational
s, the discriminant is −4. The reason for this distinction relates to general algebraic number theory. The ring of integers
of K is spanned by 1 and the square root of d only in the second case, and in the first case there are such integers that lie at half the 'lattice points' (for example, when d = −3, these are the Eisenstein integer
s, given by the complex cube roots of unity).
The set of discriminants of quadratic fields is exactly the set of fundamental discriminant
s.
OK of a quadratic field K.
In line with general theory of splitting of prime ideals in Galois extensions
, this may be
p is inert: (p) is a prime ideal
p splits: (p) is a product of two distinct prime ideals of OK.
p is ramified: (p) is the square of a prime ideal of OK.
The third case happens if and only if p divides the discriminant D. The first and second cases occur when the Kronecker symbol
(D/p) equals −1 and +1, respectively. For example, if p is an odd prime not dividing D, then p splits if and only if D is congruent to a square modulo p. The first two cases are in a certain sense equally likely to occur as p runs through the primes, see Chebotarev density theorem.
The law of quadratic reciprocity
implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D, where D is the field discriminant.
generated by a primitive p-th root of unity, with p a prime number > 2. The uniqueness is a consequence of Galois theory
, there being a unique subgroup of index
2 in the Galois group over Q. As explained at Gaussian period
, the discriminant of the quadratic field is p for p = 4n + 1 and −p for p = 4n + 3. This can also be predicted from enough ramification
theory. In fact p is the only prime that ramifies in the cyclotomic field, so that p is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants −4p and 4p in the respective cases.
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, a quadratic field is an algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
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 integer
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
s 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
Absolute value (algebra)
In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping | x | from D to the real numbers R satisfying:* | x | ≥ 0,*...
are real or complex.
Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic form
Binary quadratic form
In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variableswhere a, b, c are the coefficients...
s. There remain some unsolved problems. The class number problem is particularly important.
Discriminant
The discriminantDiscriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field...
of the quadratic field Q(√d) is d if d is congruent to 1 modulo 4, and otherwise 4d. For example, when d is −1 so that K is the field of so-called Gaussian rational
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....
s, the discriminant is −4. The reason for this distinction relates to general algebraic number theory. The ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
of K is spanned by 1 and the square root of d only in the second case, and in the first case there are such integers that lie at half the 'lattice points' (for example, when d = −3, these are the Eisenstein integer
Eisenstein integer
In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
s, given by the complex cube roots of unity).
The set of discriminants of quadratic fields is exactly the set of fundamental discriminant
Fundamental discriminant
In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of Q. Conversely, every integer D with is the discriminant of some binary quadratic form with integer...
s.
Prime factorization into ideals
Any prime number p gives rise to an ideal pOK in the ring of integersRing of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
OK of a quadratic field K.
In line with general theory of splitting of prime ideals in Galois extensions
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 OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory...
, this may be
p is inert: (p) is a prime ideal
- The quotient ring is the finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with p2 elements: OK/pOK = Fp2
p splits: (p) is a product of two distinct prime ideals of OK.
- The quotient ring is the product OK/pOK = Fp × Fp.
p is ramified: (p) is the square of a prime ideal of OK.
- The quotient ring contains non-zero nilpotentNilpotentIn mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
elements.
The third case happens if and only if p divides the discriminant D. The first and second cases occur when the Kronecker symbol
Kronecker symbol
In number theory, the Kronecker symbol, written as \left or , is a generalization of the Jacobi symbol to all integers n. It was introduced by Leopold Kronecker.-Definition:...
(D/p) equals −1 and +1, respectively. For example, if p is an odd prime not dividing D, then p splits if and only if D is congruent to a square modulo p. The first two cases are in a certain sense equally likely to occur as p runs through the primes, see Chebotarev density theorem.
The law of quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D, where D is the field discriminant.
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 fieldCyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
generated by a primitive p-th root of unity, with p a prime number > 2. The uniqueness is a consequence of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, there being a unique subgroup of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
2 in the Galois group over Q. As explained at Gaussian period
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 . They are basic in the classical theory called cyclotomy...
, the discriminant of the quadratic field is p for p = 4n + 1 and −p for p = 4n + 3. This can also be predicted from enough ramification
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
theory. In fact p is the only prime that ramifies in the cyclotomic field, so that p is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants −4p and 4p in the respective cases.
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.See also
- Class number problem
- Stark–Heegner theoremStark–Heegner theoremIn number theory, a branch of mathematics, the Stark–Heegner theorem states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers...
- Heegner numberHeegner numberIn number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q has class number 1...
- Quadratic irrationalQuadratic irrationalIn mathematics, a quadratic irrational is an irrational number that is the solution to some quadratic equation with rational coefficients...
- Quadratic integerQuadratic integerIn number theory, quadratic integers are a generalization of the rational integers to quadratic fields. Important examples include the Gaussian integers and the Eisenstein integers. Though they have been studied for more than a hundred years, many open problems remain.- Definition :Quadratic...