Inalgebraic number theory, aquadratic field is analgebraic number field ofdegree two over${\displaystyle \mathbf{Q}}$, therational numbers.

Every such quadratic field is some${\displaystyle \mathbf{Q}(\sqrt{d})}$ where${\displaystyle d}$ is a (uniquely defined)square-free integer different from${\displaystyle 0}$ and${\displaystyle 1}$. If${\displaystyle d>0}$, the corresponding quadratic field is called areal quadratic field, and, if, it is called animaginary quadratic field or acomplex quadratic field, corresponding to whether or not it is asubfield of the field of thereal numbers.

Quadratic fields have been studied in great depth, initially as part of the theory ofbinary quadratic forms. There remain some unsolved problems. Theclass number problem is particularly important.

For a nonzero square free integer${\displaystyle d}$, thediscriminant of the quadratic field${\displaystyle K=\mathbf{Q}(\sqrt{d})}$ is${\displaystyle d}$ if${\displaystyle d}$ is congruent to${\displaystyle 1}$ modulo${\displaystyle 4}$, and otherwise${\displaystyle 4d}$. For example, if${\displaystyle d}$ is${\displaystyle -1}$, then${\displaystyle K}$ is the field ofGaussian rationals and the discriminant is${\displaystyle -4}$. The reason for such a distinction is that thering of integers of${\displaystyle K}$ is generated by${\displaystyle (1+\sqrt{d})/2}$ in the first case and by${\displaystyle \sqrt{d}}$ in the second case.

The set of discriminants of quadratic fields is exactly the set offundamental discriminants (apart from${\displaystyle 1}$, which is a fundamental discriminant but not the discriminant of a quadratic field).

Any prime number${\displaystyle p}$ gives rise to an ideal${\displaystyle p{\mathcal{O}}_K}$ in thering of integers${\displaystyle {\mathcal{O}}_K}$ of a quadratic field${\displaystyle K}$. In line with general theory ofsplitting of prime ideals in Galois extensions, this may be^[1]

${\displaystyle p}$ isinert

${\displaystyle (p)}$ is a prime ideal.

The quotient ring is thefinite field with${\displaystyle p^2}$ elements:${\displaystyle {\mathcal{O}}_K/p{\mathcal{O}}_K={\mathbf{F}}_{p^2}}$.

${\displaystyle p}$splits

${\displaystyle (p)}$ is a product of two distinct prime ideals of${\displaystyle {\mathcal{O}}_K}$.

The quotient ring is the product${\displaystyle {\mathcal{O}}_K/p{\mathcal{O}}_K={\mathbf{F}}_p\times {\mathbf{F}}_p}$.

${\displaystyle p}$ isramified

${\displaystyle (p)}$ is the square of a prime ideal of${\displaystyle {\mathcal{O}}_K}$.

The quotient ring contains non-zeronilpotent elements.

The third case happens if and only if${\displaystyle p}$ divides the discriminant${\displaystyle D}$. The first and second cases occur when theKronecker symbol${\displaystyle (D/p)}$ equals${\displaystyle -1}$ and${\displaystyle +1}$, respectively. For example, if${\displaystyle p}$ is an odd prime not dividing${\displaystyle D}$, then${\displaystyle p}$ splits if and only if${\displaystyle D}$ is congruent to a square modulo${\displaystyle p}$. The first two cases are, in a certain sense, equally likely to occur as${\displaystyle p}$ runs through the primes—seeChebotarev density theorem.^[2]

The law ofquadratic reciprocity implies that the splitting behaviour of a prime${\displaystyle p}$ in a quadratic field depends only on${\displaystyle p}$ modulo${\displaystyle D}$, where${\displaystyle D}$ is the field discriminant.

Determining the class group of a quadratic field extension can be accomplished usingMinkowski's bound and theKronecker symbol because of the finiteness of theclass group.^[3] A quadratic field${\displaystyle K=\mathbf{Q}(\sqrt{d})}$ hasdiscriminantso the Minkowski bound is^[4]

Then, the ideal class group is generated by the prime ideals whose norm is less than${\displaystyle M_K}$. This can be done by looking at the decomposition of the ideals${\displaystyle (p)}$ for${\displaystyle p\in \mathbf{Z}}$ prime where^{[1]page 72} These decompositions can be found using theDedekind–Kummer theorem.

A classical example of the construction of a quadratic field is to take the unique quadratic field inside thecyclotomic field generated by a primitive${\displaystyle p}$th root of unity, with${\displaystyle p}$ an odd prime number. The uniqueness is a consequence ofGalois theory, there being a unique subgroup ofindex${\displaystyle 2}$ in the Galois group over${\displaystyle \mathbf{Q}}$. As explained atGaussian period, the discriminant of the quadratic field is${\displaystyle p}$ for${\displaystyle p=4n+1}$ and${\displaystyle -p}$ for${\displaystyle p=4n+3}$. This can also be predicted from enoughramification theory. In fact,${\displaystyle p}$ is the only prime that ramifies in the cyclotomic field, so${\displaystyle p}$ is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants${\displaystyle -4p}$ and${\displaystyle 4p}$ in the respective cases.

If one takes the other cyclotomic fields, they have Galois groups with extra${\displaystyle 2}$-torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant${\displaystyle D}$ can be obtained as a subfield of a cyclotomic field of${\displaystyle D}$-th roots of unity. This expresses the fact that theconductor of a quadratic field is the absolute value of its discriminant, a special case of theconductor-discriminant formula.

The following table shows someorders of small discriminant of quadratic fields. Themaximal order of an algebraic number field is itsring 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 ofDiscriminant of an algebraic number field § Definition.

For real quadratic integer rings, theideal class number, which measures the failure of unique factorization, is given inOEIS A003649; for the imaginary case, they are given inOEIS A000924.

Order	Discriminant	Class number	Units	Comments
${\displaystyle \mathbf{Z}\left[\sqrt{-5}\right]}$	${\displaystyle -20}$	${\displaystyle 2}$	${\displaystyle \pm 1}$	Ideal classes${\displaystyle (1)}$,${\displaystyle (2,1+\sqrt{-5})}$
${\displaystyle \mathbf{Z}\left[(1+\sqrt{-19})/2\right]}$	${\displaystyle -19}$	${\displaystyle 1}$	${\displaystyle \pm 1}$	Principal ideal domain, notEuclidean
${\displaystyle \mathbf{Z}\left[2\sqrt{-1}\right]}$	${\displaystyle -16}$	${\displaystyle 1}$	${\displaystyle \pm 1}$	Non-maximal order
${\displaystyle \mathbf{Z}\left[(1+\sqrt{-15})/2\right]}$	${\displaystyle -15}$	${\displaystyle 2}$	${\displaystyle \pm 1}$	Ideal classes${\displaystyle (1)}$,${\displaystyle (1,(1+\sqrt{-15})/2)}$
${\displaystyle \mathbf{Z}\left[\sqrt{-3}\right]}$	${\displaystyle -12}$	${\displaystyle 1}$	${\displaystyle \pm 1}$	Non-maximal order
${\displaystyle \mathbf{Z}\left[(1+\sqrt{-11})/2\right]}$	${\displaystyle -11}$	${\displaystyle 1}$	${\displaystyle \pm 1}$	Euclidean
${\displaystyle \mathbf{Z}\left[\sqrt{-2}\right]}$	${\displaystyle -8}$	${\displaystyle 1}$	${\displaystyle \pm 1}$	Euclidean
${\displaystyle \mathbf{Z}\left[(1+\sqrt{-7})/2\right]}$	${\displaystyle -7}$	${\displaystyle 1}$	${\displaystyle \pm 1}$	Kleinian integers
${\displaystyle \mathbf{Z}\left[\sqrt{-1}\right]}$	${\displaystyle -4}$	${\displaystyle 1}$	${\displaystyle \pm 1,\pm i}$ (cyclic of order${\displaystyle 4}$)	Gaussian integers
${\displaystyle \mathbf{Z}\left[(1+\sqrt{-3})/2\right]}$	${\displaystyle -3}$	${\displaystyle 1}$	${\displaystyle \pm 1,(\pm 1\pm \sqrt{-3})/2}$.	Eisenstein integers
${\displaystyle \mathbf{Z}\left[\sqrt{-21}\right]}$	${\displaystyle -84}$	${\displaystyle 4}$		Class group non-cyclic:${\displaystyle (\mathbf{Z}/2\mathbf{Z}{)}^2}$
${\displaystyle \mathbf{Z}\left[(1+\sqrt{5})/2\right]}$	${\displaystyle 5}$	${\displaystyle 1}$	${\displaystyle \pm ((1+\sqrt{5})/2{)}^n}$ (norm${\displaystyle (-1{)}^n}$)
${\displaystyle \mathbf{Z}\left[\sqrt{2}\right]}$	${\displaystyle 8}$	${\displaystyle 1}$	${\displaystyle \pm (1+\sqrt{2}{)}^n}$ (norm${\displaystyle (-1{)}^n}$)
${\displaystyle \mathbf{Z}\left[\sqrt{3}\right]}$	${\displaystyle 12}$	${\displaystyle 1}$	${\displaystyle \pm (2+\sqrt{3}{)}^n}$ (norm${\displaystyle 1}$)
${\displaystyle \mathbf{Z}\left[(1+\sqrt{13})/2\right]}$	${\displaystyle 13}$	${\displaystyle 1}$	${\displaystyle \pm ((3+\sqrt{13})/2{)}^n}$ (norm${\displaystyle (-1{)}^n}$)
${\displaystyle \mathbf{Z}\left[(1+\sqrt{17})/2\right]}$	${\displaystyle 17}$	${\displaystyle 1}$	${\displaystyle \pm (4+\sqrt{17}{)}^n}$ (norm${\displaystyle (-1{)}^n}$)
${\displaystyle \mathbf{Z}\left[\sqrt{5}\right]}$	${\displaystyle 20}$	${\displaystyle 1}$	${\displaystyle \pm (\sqrt{5}+2{)}^n}$ (norm${\displaystyle (-1{)}^n}$)	Non-maximal order

Some of these examples are listed in Artin,Algebra (2nd ed.), §13.8.

• Buell, Duncan (1989),Binary quadratic forms: classical theory and modern computations,Springer-Verlag,ISBN 0-387-97037-1 Chapter 6.
• Samuel, Pierre (1972),Algebraic Theory of Numbers (Hardcover ed.), Paris / Boston: Hermann / Houghton Mifflin Company,ISBN 978-0-901-66506-5
  • Samuel, Pierre (2008),Algebraic Theory of Numbers (Paperback ed.), Dover,ISBN 978-0-486-46666-8
• Stewart, I. N.; Tall, D. O. (1979),Algebraic number theory, Chapman and Hall,ISBN 0-412-13840-9 Chapter 3.1.

• Weisstein, Eric W."Quadratic Field".MathWorld.
• "Quadratic field",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Algebraic number theory
• Field (mathematics)

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