Editing Quadratic field

{{short description|Field (mathematics) generated by the square root of an integer}}
In [[algebraic number theory]], a '''quadratic field''' is an [[algebraic number field]] of [[Degree of a field extension|degree]] two over <math>\mathbf{Q}</math>, the [[rational number]]s.

Every such quadratic field is some <math>\mathbf{Q}(\sqrt{d})</math> where <math>d</math> is a (uniquely defined) [[square-free integer]] different from <math>0</math> and <math>1</math>. If <math>d>0</math>, the corresponding quadratic field is called a '''real quadratic field''', and, if <math>d<0</math>, it is called an '''imaginary quadratic field''' or a '''complex quadratic field''', corresponding to whether or not it is a [[Field extension|subfield]] of the field of the [[real number]]s.

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.

==Ring of integers==
{{Main|Quadratic integer}}

==Discriminant==
For a nonzero square free integer <math>d</math>, the [[Discriminant of an algebraic number field|discriminant]] of the quadratic field <math>K = \mathbf{Q}(\sqrt{d})</math> is <math>d</math> if <math>d</math> is congruent to <math>1</math> modulo <math>4</math>, and otherwise <math>4d</math>. For example, if <math>d</math> is <math>-1</math>, then <math>K</math> is the field of [[Gaussian rational]]s and the discriminant is <math>-4</math>. The reason for such a distinction is that the [[ring of integers]] of <math>K</math> is generated by <math>(1+\sqrt{d})/2</math> in the first case and by <math>\sqrt{d}</math> in the second case.

The set of discriminants of quadratic fields is exactly the set of [[fundamental discriminant]]s (apart from <math>1</math>, which is a fundamental discriminant but not the discriminant of a quadratic field).

==Prime factorization into ideals==
Any prime number <math>p</math> gives rise to an ideal <math>p\mathcal{O}_K</math> in the [[ring of integers]] <math>\mathcal{O}_K</math> of a quadratic field <math>K</math>. In line with general theory of [[splitting of prime ideals in Galois extensions]], this may be<ref name=":0">{{Cite web|title=Number Rings|url=http://websites.math.leidenuniv.nl/algebra/ant.pdf|last=Stevenhagen|pages=36}}</ref>

;<math>p</math> is '''inert''': <math>(p)</math> is a prime ideal.
: The quotient ring is the [[finite field]] with <math>p^2</math> elements: <math>\mathcal{O}_K / p\mathcal{O}_K = \mathbf{F}_{p^2}</math>.
;<math>p</math> '''splits''': <math>(p)</math> is a product of two distinct prime ideals of <math>\mathcal{O}_K</math>.
: The quotient ring is the product <math>\mathcal{O}_K/p\mathcal{O}_K = \mathbf{F}_p\times\mathbf{F}_p</math>.
;<math>p</math> is '''ramified''': <math>(p)</math> is the square of a prime ideal of <math>\mathcal{O}_K</math>.
:The quotient ring contains non-zero [[nilpotent]] elements.

The third case happens if and only if <math>p</math> divides the discriminant <math>D</math>.  The first and second cases occur when the [[Kronecker symbol]] <math>(D/p)</math> equals <math>-1</math> and <math>+1</math>, respectively. For example, if <math>p</math> is an odd prime not dividing <math>D</math>, then <math>p</math> splits if and only if <math>D</math> is congruent to a square modulo <math>p</math>.  The first two cases are, in a certain sense, equally likely to occur as <math>p</math> runs through the primes—see [[Chebotarev density theorem]].<ref>{{Harvnb|Samuel|1972|pp=76f}}</ref>

The law of [[quadratic reciprocity]] implies that the splitting behaviour of a prime <math>p</math> in a quadratic field depends only on <math>p</math> modulo <math>D</math>, where <math>D</math> is the field discriminant.

== Class group ==
Determining the class group of a quadratic field extension can be accomplished using [[Minkowski's bound]] and the [[Kronecker symbol]] because of the finiteness of the [[Ideal class group|class group]].<ref>{{Cite web|title=Algebraic Number Theory, A Computational Approach|url=https://wstein.org/books/ant/ant.pdf|last=Stein|first=William|pages=77–86}}</ref> A quadratic field <math>K = \mathbf{Q}(\sqrt{d})</math> has [[Discriminant of an algebraic number field|discriminant]]
<math display=block>\Delta_K = \begin{cases}
d & d \equiv 1 \pmod 4 \\
4d & d \equiv 2,3 \pmod 4;
\end{cases}</math>
so the Minkowski bound is<ref>{{Cite web|last=Conrad|first=Keith|title=CLASS GROUP CALCULATIONS|url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgpex.pdf}}</ref><math display=block>M_K = \begin{cases}
2\sqrt{|\Delta|}/\pi & d < 0 \\
\sqrt{|\Delta|}/2 & d > 0 .
\end{cases}
 </math>

Then, the ideal class group is generated by the prime ideals whose norm is less than <math>M_K</math>. This can be done by looking at the decomposition of the ideals <math>(p)</math> for <math>p \in \mathbf{Z}</math> prime where <math>|p| < M_k.</math><ref name=":0" /> <sup>page 72</sup> These decompositions can be found using the [[Dedekind–Kummer theorem]].

==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]] generated by a primitive <math>p</math>th root of unity, with <math>p</math> an odd prime number. The uniqueness is a consequence of [[Galois theory]], there being a unique subgroup of [[Index of a subgroup|index]] <math>2</math> in the Galois group over <math>\mathbf{Q}</math>. As explained at [[Gaussian period]], the discriminant of the quadratic field is <math>p</math> for <math>p=4n+1</math> and <math>-p</math> for <math>p=4n+3</math>. This can also be predicted from enough [[Ramification (mathematics)|ramification]] theory. In fact, <math>p</math> is the only prime that ramifies in the cyclotomic field, so <math>p</math> is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants <math>-4p</math> and <math>4p</math> in the respective cases.

===Other cyclotomic fields===
If one takes the other cyclotomic fields, they have Galois groups with extra <math>2</math>-torsion, so contain at least three quadratic fields.  In general a quadratic field of field discriminant <math>D</math> can be obtained as a subfield of a cyclotomic field of <math>D</math>-th roots of unity.  This expresses the fact that the [[Conductor (class field theory)|conductor]] of a quadratic field is the absolute value of its discriminant, a special case of the [[conductor-discriminant formula]].

==Orders of quadratic number fields of small discriminant==

The following table shows some [[order (ring theory)|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 {{slink|Discriminant of an algebraic number field|Definition}}.

For real quadratic integer rings, the [[Ideal class group#Properties|ideal class number]], which measures the failure of unique factorization, is given in [https://oeis.org/A003649 OEIS A003649]; for the imaginary case, they are given in [https://oeis.org/A000924 OEIS A000924].

{| class="wikitable"
|-
! Order
! Discriminant
! Class number
! Units
! Comments
|-
| <math>\mathbf{Z}\left[\sqrt{-5}\right]</math>
| <math>-20</math>
| <math>2</math>
| <math>\pm 1</math>
| Ideal classes <math>(1)</math>, <math>(2,1+\sqrt{-5})</math>
|-
| <math>\mathbf{Z}\left[(1+\sqrt{-19})/2\right]</math>
| <math>-19</math>
| <math>1</math>
| <math>\pm 1</math>
| [[Principal ideal domain]], not [[Euclidean domain|Euclidean]]
|-
| <math>\mathbf{Z}\left[2\sqrt{-1}\right]</math>
| <math>-16</math>
| <math>1</math>
| <math>\pm 1</math>
|Non-maximal order
|-
| <math>\mathbf{Z}\left[(1+\sqrt{-15})/2\right]</math>
| <math>-15</math>
| <math>2</math>
| <math>\pm 1</math>
|Ideal classes <math>(1)</math>, <math>(1,(1+\sqrt{-15})/2)</math>
|-
| <math>\mathbf{Z}\left[\sqrt{-3}\right]</math>
| <math>-12</math>
| <math>1</math>
| <math>\pm 1</math>
|Non-maximal order
|-
| <math>\mathbf{Z}\left[(1+\sqrt{-11})/2\right]</math>
| <math>-11</math>
| <math>1</math>
| <math>\pm 1</math>
| Euclidean
|-
| <math>\mathbf{Z}\left[\sqrt{-2}\right]</math>
| <math>-8</math>
| <math>1</math>
| <math>\pm 1</math>
| Euclidean
|-
| <math>\mathbf{Z}\left[(1+\sqrt{-7})/2\right]</math>
| <math>-7</math>
| <math>1</math>
| <math>\pm 1</math>
|[[Kleinian integers]]
|-
| <math>\mathbf{Z}\left[\sqrt{-1}\right]</math>
| <math>-4</math>
| <math>1</math>
| <math>\pm 1,\pm i</math> (cyclic of order <math>4</math>)
|[[Gaussian integers]]
|-
| <math>\mathbf{Z}\left[(1+\sqrt{-3})/2\right]</math>
| <math>-3</math>
| <math>1</math>
| <math>\pm 1,(\pm 1 \pm \sqrt{-3})/2</math>.
|[[Eisenstein integers]]
|-
| <math>\mathbf{Z}\left[ \sqrt{-21}\right]</math>
| <math>-84</math>
| <math>4</math>
| 
|Class group non-cyclic: <math>(\mathbf{Z}/2\mathbf{Z})^2</math>
|-
| <math>\mathbf{Z}\left[ (1+\sqrt{5})/2\right]</math>
| <math>5</math>
| <math>1</math>
| <math>\pm((1+\sqrt{5})/2)^n</math> (norm <math>(-1)^n</math>)
|
|-
| <math>\mathbf{Z}\left[ \sqrt{2}\right]</math>
| <math>8</math>
| <math>1</math>
| <math>\pm(1+\sqrt{2})^n</math> (norm <math>(-1)^n</math>)
|
|-
| <math>\mathbf{Z}\left[ \sqrt{3}\right]</math>
| <math>12</math>
| <math>1</math>
| <math>\pm(2+\sqrt{3})^n</math> (norm <math>1</math>)
|
|-
| <math>\mathbf{Z}\left[ (1+\sqrt{13})/2\right]</math>
| <math>13</math>
| <math>1</math>
| <math>\pm((3+\sqrt{13})/2)^n</math> (norm <math>(-1)^n</math>)
|
|-
| <math>\mathbf{Z}\left[ (1+\sqrt{17})/2\right]</math>
| <math>17</math>
| <math>1</math>
| <math>\pm(4+\sqrt{17})^n</math> (norm <math>(-1)^n</math>)
|
|-
| <math>\mathbf{Z}\left[\sqrt{5}\right]</math>
| <math>20</math>
| <math>1</math>
| <math>\pm(\sqrt{5}+2)^n</math> (norm <math>(-1)^n</math>)
|Non-maximal order
|}

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

==See also==
{{Div col}}
*[[Eisenstein–Kronecker number]]
*[[Genus character]]
*[[Heegner number]]
*[[Infrastructure (number theory)]]
*[[Quadratic integer]]
*[[Quadratic irrational]]
*[[Stark–Heegner theorem]]
*[[Dedekind zeta function]]
*[[Quadratically closed field]]
{{Div col end}}

==Notes==
{{reflist}}

==References==
* {{Citation |last=Buell |first=Duncan |title=Binary quadratic forms: classical theory and modern computations |publisher=[[Springer-Verlag]] |year=1989 |isbn=0-387-97037-1 |url-access=registration |url=https://archive.org/details/binaryquadraticf0000buel }} Chapter 6.
* {{Citation|last=Samuel|first=Pierre|author-link=Pierre Samuel|year=1972|title=Algebraic Theory of Numbers|publisher=Hermann / Houghton Mifflin Company|location=Paris / Boston|edition=Hardcover|isbn=978-0-901-66506-5}}
** {{Citation|last=Samuel|first=Pierre|year=2008|title=Algebraic Theory of Numbers|publisher=Dover|edition=Paperback|isbn=978-0-486-46666-8|url={{google books|u9vCAgAAQBAJ|plainurl=yes}}}}
* {{Citation |last1=Stewart |first1=I. N. |author-link1=Ian Stewart (mathematician) |last2=Tall |first2=D. O. |title=Algebraic number theory |publisher=Chapman and Hall |year=1979 |isbn=0-412-13840-9}} Chapter 3.1.

== External links ==

*{{MathWorld|title=Quadratic Field|id=QuadraticField}}
*{{springerEOM|title=Quadratic field|id=Quadratic_field&oldid=25501}}

[[Category:Algebraic number theory]]
[[Category:Field (mathematics)]]