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Home > Quadratic field

In mathematics, a quadratic field is a field extension K/Q of the form

where d is a non-zero rational number. Such extensions run over all field extensions of the rational number field that are of degree 2 ( quadratic extensions). If d > 0 this is called a real quadratic field, and for d < 0 an imaginary quadratic field. Such fields are a basic class of examples in algebraic number theory. They have been studied in great depth, initially as part of the theory of quadratic forms. There remain some unsolved problems.

We can take d to be an integer. In fact d can be changed by any perfect square, so that to get all the fields exactly once we should take representative d up to squares. That is, we may take d to be a square free integer, positive or negative. The discriminant of the corresponding quadratic field is then 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 rationals, the discriminant is −4. The reason for the definition relates to general algebraic number theory; the algebraic integers in K are 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 the complex cube roots of unity).

1 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 p-th root of unity, with p a prime number > 2. The uniqueness is a consequence of Galois theoryIn mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. In other words, the Galois theory is the study of solutions to polynomials and how the different solutions are related to each other, there being a unique subgroup of index 2 in the Galois group over Q. As explained at Gaussian periodIn mathematics, a Gaussian period is a certain kind of sum of roots of unity. They permit explicit calculations in cyclotomic fields, in relation both with Galois theory and with harmonic analysis ( discrete Fourier transform). They are basic in the class, 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 ramificationAlgebraic number theory Algebraic topology Complex analysis 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. It is 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' disciminants −4p and 4p in the respective cases.

If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields.

2 Prime factorization into ideals

Any prime number p gives rise to an ideal p.O_K in the ring of integers O_K of a quadratic field K. In line with general theory, this may be

1. a prime ideal, or
2. a product of two distinct prime ideals of O_K, or
3. the square of a prime ideal of O_K.

The third case happens only for the primes dividing the discriminant. The other two cases both occur, as p varies, and in a certain sense are equally likely. The second case is called p splitting, and the first case p remaining inert. The first case corresponds to the quotient ring

O_K/p.O_K

being a finite fieldIn abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theo with p² elements, the second to it being a product of two finite fields each with p elements. In the third case, p is said to ramify, and the quotient ring contains non-zero nilpotentRing theory In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that x n 0. Examples This definition can be applied in particular to square matrices. The matrix : is nilpotent because A''3 0. It can be elements.

See also:

• Stark-Heegner theoremThe Stark-Heegner theorem is a theorem in number theory that tells precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers. It solves a special case of Gauss's class number problem of determining the number o, Heegner numberIn number theory, a Heegner number is a positive integer n such that the quadratic field Q √ n has class number 1. Equivalently, its ring of integers has unique factorization. See Stark-Heegner theorem for the proof that such n that are square free
• Quadratic irrational

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Topics: Quadratic Field Prime Number Theory Case Discriminant Galois...

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