Quadratic field

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In algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers.

Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and, if , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.

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

Ring of integers

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Discriminant

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For a nonzero square free integer  , the discriminant of the quadratic field   is   if   is congruent to   modulo  , and otherwise  . For example, if   is  , then   is the field of Gaussian rationals and the discriminant is  . The reason for such a distinction is that the ring of integers of   is generated by   in the first case and by   in the second case.

The set of discriminants of quadratic fields is exactly the set of fundamental discriminants (apart from  , which is a fundamental discriminant but not the discriminant of a quadratic field).

Prime factorization into ideals

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Any prime number   gives rise to an ideal   in the ring of integers   of a quadratic field  . In line with general theory of splitting of prime ideals in Galois extensions, this may be[1]

  is inert
  is a prime ideal.
The quotient ring is the finite field with   elements:  .
  splits
  is a product of two distinct prime ideals of  .
The quotient ring is the product  .
  is ramified
  is the square of a prime ideal of  .
The quotient ring contains non-zero nilpotent elements.

The third case happens if and only if   divides the discriminant  . The first and second cases occur when the Kronecker symbol   equals   and  , respectively. For example, if   is an odd prime not dividing  , then   splits if and only if   is congruent to a square modulo  . The first two cases are, in a certain sense, equally likely to occur as   runs through the primes—see Chebotarev density theorem.[2]

The law of quadratic reciprocity implies that the splitting behaviour of a prime   in a quadratic field depends only on   modulo  , where   is the field discriminant.

Class group

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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 class group.[3] A quadratic field   has discriminant   so the Minkowski bound is[4] 

Then, the ideal class group is generated by the prime ideals whose norm is less than  . This can be done by looking at the decomposition of the ideals   for   prime where  [1] page 72 These decompositions can be found using the Dedekind–Kummer theorem.

Quadratic subfields of cyclotomic fields

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The quadratic subfield of the prime cyclotomic field

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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  th root of unity, with   an odd prime number. The uniqueness is a consequence of Galois theory, there being a unique subgroup of index   in the Galois group over  . As explained at Gaussian period, the discriminant of the quadratic field is   for   and   for  . This can also be predicted from enough ramification theory. In fact,   is the only prime that ramifies in the cyclotomic field, so   is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants   and   in the respective cases.

Other cyclotomic fields

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If one takes the other cyclotomic fields, they have Galois groups with extra  -torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant   can be obtained as a subfield of a cyclotomic field of  -th roots of unity. This expresses the fact that the 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

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The following table shows some 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 Discriminant of an algebraic number field § Definition.

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

Order Discriminant Class number Units Comments
        Ideal classes  ,  
        Principal ideal domain, not Euclidean
        Non-maximal order
        Ideal classes  ,  
        Non-maximal order
        Euclidean
        Euclidean
        Kleinian integers
        (cyclic of order  ) Gaussian integers
       . Eisenstein integers
      Class group non-cyclic:  
        (norm  )
        (norm  )
        (norm  )
        (norm  )
        (norm  )
        (norm  ) Non-maximal order

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

See also

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Notes

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  1. ^ a b Stevenhagen. "Number Rings" (PDF). p. 36.
  2. ^ Samuel 1972, pp. 76f
  3. ^ Stein, William. "Algebraic Number Theory, A Computational Approach" (PDF). pp. 77–86.
  4. ^ Conrad, Keith. "CLASS GROUP CALCULATIONS" (PDF).

References

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