Quadratic field

In algebraic number theory, a quadratic field is a special kind of number system that builds on the rational numbers. It is an algebraic number field where we add the square root of a special number to the regular numbers we already know. This special number, called d, must be a whole number that cannot be written as a perfect square and cannot be 0 or 1.

Every quadratic field looks like this: we take all the numbers we can make by adding the square root of d to the rational numbers. This creates a new number field that has exactly two copies, or "dimensions," more than the rational numbers alone. That is why we say the quadratic field has a degree of two.

If the number d is positive, the quadratic field is called a real quadratic field because it includes real numbers we can see on a number line. If d is negative, the field includes numbers called imaginary numbers, which help us solve problems that need the square root of negative numbers. Quadratic fields are important in number theory because they help us understand the properties of numbers and solve equations that cannot be solved using just the rational numbers.

Discriminant

For a special number called a quadratic field, there is something called a discriminant. This helps us understand the field better.

If the number is a special type (it ends in 1 when divided by 4), the discriminant is the number itself. Otherwise, it is four times the number. For example, if the number is -1, the discriminant becomes -4. This helps mathematicians study these number fields more easily.

Prime factorization into ideals

When we look at prime numbers in special number systems called quadratic fields, they can behave in different ways. A prime number can stay as it is, split into two smaller primes, or change in a special way called ramifying. These changes depend on the properties of the quadratic field and the prime number itself. The way a prime number behaves can often be predicted by looking at the field’s discriminant and using rules from number theory.

Class group

Determining the class group of a quadratic field can be done using special rules and symbols because the group has a limited number of parts. A quadratic field is a special type of number system that includes square roots of certain numbers.

The way these fields are built helps us understand their properties better. We can study small parts of these fields to learn more about their structure. This is done by looking at how certain numbers break apart, following specific math rules.

Quadratic subfields of cyclotomic fields

A quadratic field can be found inside a special kind of number field called a cyclotomic field. This happens when we use a certain type of number related to a prime number p. Because of how these number fields are built, there is only one quadratic field inside each of these cyclotomic fields.

Other cyclotomic fields can contain more than one quadratic field. In fact, they often have at least three. Any quadratic field can be seen as part of a cyclotomic field built from roots of unity linked to the field’s discriminant. This shows a deep connection between the structure of these fields and their discriminants.

cyclotomic field Galois theory index Gaussian period ramification conductor-discriminant formula

Orders of quadratic number fields of small discriminant

This table shows some special orders of small size in quadratic fields. These orders are connected to important numbers called discriminants, which help describe the properties of these fields.

The table also includes numbers that show how well these fields can break down into simpler parts, called ideal class numbers.

Order	Discriminant	Class number	Units	Comments
Z [ − 5 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-5}}\right]}	− 20 {\displaystyle -20}	2 {\displaystyle 2}	± 1 {\displaystyle \pm 1}	Ideal classes ( 1 ) {\displaystyle (1)} , ( 2 , 1 + − 5 ) {\displaystyle (2,1+{\sqrt {-5}})}
Z [ 1 2 ( 1 + − 19 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {-19}})\right]}	− 19 {\displaystyle -19}	1 {\displaystyle 1}	± 1 {\displaystyle \pm 1}	Principal ideal domain, not Euclidean
Z [ 2 − 1 ] {\displaystyle \mathbf {Z} \left[2{\sqrt {-1}}\right]}	− 16 {\displaystyle -16}	1 {\displaystyle 1}	± 1 {\displaystyle \pm 1}	Non-maximal order
Z [ 1 2 ( 1 + − 15 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {-15}})\right]}	− 15 {\displaystyle -15}	2 {\displaystyle 2}	± 1 {\displaystyle \pm 1}	Ideal classes ( 1 ) {\displaystyle (1)} , ( 1 , 1 2 ( 1 + − 15 ) ) {\displaystyle \left(1,{\tfrac {1}{2}}(1+{\sqrt {-15}})\right)}
Z [ − 3 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-3}}\right]}	− 12 {\displaystyle -12}	1 {\displaystyle 1}	± 1 {\displaystyle \pm 1}	Non-maximal order
Z [ 1 2 ( 1 + − 11 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {-11}})\right]}	− 11 {\displaystyle -11}	1 {\displaystyle 1}	± 1 {\displaystyle \pm 1}	Euclidean
Z [ − 2 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-2}}\right]}	− 8 {\displaystyle -8}	1 {\displaystyle 1}	± 1 {\displaystyle \pm 1}	Euclidean
Z [ 1 2 ( 1 + − 7 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {-7}})\right]}	− 7 {\displaystyle -7}	1 {\displaystyle 1}	± 1 {\displaystyle \pm 1}	Kleinian integers
Z [ − 1 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-1}}\right]}	− 4 {\displaystyle -4}	1 {\displaystyle 1}	± 1 , ± i {\displaystyle \pm 1,\pm i} (cyclic of order 4 {\displaystyle 4} )	Gaussian integers
Z [ 1 2 ( 1 + − 3 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {-3}})\right]}	− 3 {\displaystyle -3}	1 {\displaystyle 1}	± 1 , 1 2 ( ± 1 ± − 3 ) {\displaystyle \pm 1,{\tfrac {1}{2}}(\pm 1\pm {\sqrt {-3}})} (cyclic of order 6 {\displaystyle 6} )	Eisenstein integers
Z [ − 21 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-21}}\right]}	− 84 {\displaystyle -84}	4 {\displaystyle 4}		Class group non-cyclic: ( Z / 2 Z ) 2 {\displaystyle (\mathbf {Z} /2\mathbf {Z} )^{2}}
Z [ 1 2 ( 1 + 5 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {5}})\right]}	5 {\displaystyle 5}	1 {\displaystyle 1}	± ( 1 2 ( 1 + 5 ) ) n {\displaystyle \pm \left({\tfrac {1}{2}}(1+{\sqrt {5}})\right)^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )	Golden integers
Z [ 2 ] {\displaystyle \mathbf {Z} \left[{\sqrt {2}}\right]}	8 {\displaystyle 8}	1 {\displaystyle 1}	± ( 1 + 2 ) n {\displaystyle \pm (1+{\sqrt {2}})^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )
Z [ 3 ] {\displaystyle \mathbf {Z} \left[{\sqrt {3}}\right]}	12 {\displaystyle 12}	1 {\displaystyle 1}	± ( 2 + 3 ) n {\displaystyle \pm (2+{\sqrt {3}})^{n}} (norm 1 {\displaystyle 1} )
Z [ 1 2 ( 1 + 13 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {13}})\right]}	13 {\displaystyle 13}	1 {\displaystyle 1}	± ( 1 2 ( 3 + 13 ) ) n {\displaystyle \pm \left({\tfrac {1}{2}}(3+{\sqrt {13}})\right)^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )
Z [ 1 2 ( 1 + 17 ) ] {\displaystyle \mathbf {Z} \left[{\tfrac {1}{2}}(1+{\sqrt {17}})\right]}	17 {\displaystyle 17}	1 {\displaystyle 1}	± ( 4 + 17 ) n {\displaystyle \pm (4+{\sqrt {17}})^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )
Z [ 5 ] {\displaystyle \mathbf {Z} \left[{\sqrt {5}}\right]}	20 {\displaystyle 20}	1 {\displaystyle 1}	± ( 5 + 2 ) n {\displaystyle \pm ({\sqrt {5}}+2)^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )	Non-maximal order