Quadratic field

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

Related Topics:
Mathematics - Field extension

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:K=mathbb{Q}(sqrt{d})

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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.

Related Topics:
Rational number - Degree - Quadratic extension - Algebraic number theory - Quadratic form

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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.

Related Topics:
Perfect square - Up to - Square free - Discriminant

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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).

Related Topics:
Gaussian rational - Algebraic integer

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