Editing Quadratic field (section)

==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.