Gauss composition law

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In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae,[1] a textbook on number theory published in 1801, in Articles 234 - 244. Gauss composition law is one of the deepest results in the theory of IBQFs and Gauss's formulation of the law and the proofs its properties as given by Gauss are generally considered highly complicated and very difficult.[2] Several later mathematicians have simplified the formulation of the composition law and have presented it in a format suitable for numerical computations. The concept has also found generalisations in several directions.

Integral binary quadratic forms

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An expression of the form  , where   are all integers, is called an integral binary quadratic form (IBQF). The form   is called a primitive IBQF if   are relatively prime. The quantity   is called the discriminant of the IBQF  . An integer   is the discriminant of some IBQF if and only if  .   is called a fundamental discriminant if and only if one of the following statements holds

  •   and is square-free,
  •   where   and   is square-free.

If   and   then   is said to be positive definite; if   and   then   is said to be negative definite; if   then   is said to be indefinite.

Equivalence of IBQFs

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Two IBQFs   and   are said to be equivalent (or, properly equivalent) if there exist integers α, β, γ, δ such that

  and  

The notation   is used to denote the fact that the two forms are equivalent. The relation " " is an equivalence relation in the set of all IBQFs. The equivalence class to which the IBQF   belongs is denoted by  .

Two IBQFs   and   are said to be improperly equivalent if

  and  

The relation in the set of IBQFs of being improperly equivalent is also an equivalence relation.

It can be easily seen that equivalent IBQFs (properly or improperly) have the same discriminant.

Gauss's formulation of the composition law

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Historical context

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The following identity, called Brahmagupta identity, was known to the Indian mathematician Brahmagupta (598–668) who used it to calculate successively better fractional approximations to square roots of positive integers:

 

Writing   this identity can be put in the form

  where  .

Gauss's composition law of IBQFs generalises this identity to an identity of the form   where   are all IBQFs and   are linear combinations of the products  .

The composition law of IBQFs

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Consider the following IBQFs:

 
 
 

If it is possible to find integers   and   such that the following six numbers

 

have no common divisors other than ±1, and such that if we let

 
 

the following relation is identically satisfied

 ,

then the form   is said to be a composite of the forms   and  . It may be noted that the composite of two IBQFs, if it exists, is not unique.

Example

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Consider the following binary quadratic forms:

 
 
 

Let

 

We have

 .

These six numbers have no common divisors other than ±1. Let

 ,
 .

Then it can be verified that

 .

Hence   is a composite of   and  .

An algorithm to find the composite of two IBQFs

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The following algorithm can be used to compute the composite of two IBQFs.[3]

Algorithm

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Given the following IBQFs having the same discriminant  :

 
 
 
  1. Compute  
  2. Compute  
  3. Compute   such that  
  4. Compute  
  5. Compute  
  6. Compute  
  7. Compute  
  8. Compute
 
 

Then   so that   is a composite of   and  .

Properties of the composition law

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Existence of the composite

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The composite of two IBQFs exists if and only if they have the same discriminant.

Equivalent forms and the composition law

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Let   be IBQFs and let there be the following equivalences:

 
 

If   is a composite of   and  , and   is a composite of   and  , then

 

A binary operation

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Let   be a fixed integer and consider set   of all possible primitive IBQFs of discriminant  . Let   be the set of equivalence classes in this set under the equivalence relation " ". Let   and   be two elements of  . Let   be a composite of the IBQFs   and   in  . Then the following equation

 

defines a well-defined binary operation " " in  .

The group GD

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  • The set   is a finite abelian group under the binary operation  .
  • The identity element in the group   =  
  • The inverse of   in   is  .

Modern approach to the composition law

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The following sketch of the modern approach to the composition law of IBQFs is based on a monograph by Duncan A. Buell.[4] The book may be consulted for further details and for proofs of all the statements made hereunder.

Quadratic algebraic numbers and integers

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Let   be the set of integers. Hereafter, in this section, elements of   will be referred as rational integers to distinguish them from algebraic integers to be defined below.

A complex number   is called a quadratic algebraic number if it satisfies an equation of the form

  where  .

  is called a quadratic algebraic integer if it satisfies an equation of the form

  where  

The quadratic algebraic numbers are numbers of the form

  where   and   has no square factors other than  .

The integer   is called the radicand of the algebraic integer  . The norm of the quadratic algebraic number   is defined as

 .

Let   be the field of rational numbers. The smallest field containing   and a quadratic algebraic number   is the quadratic field containing   and is denoted by  . This field can be shown to be

 

The discriminant   of the field   is defined by

 

Let   be a rational integer without square factors (except 1). The set of quadratic algebraic integers of radicand   is denoted by  . This set is given by

 

  is a ring under ordinary addition and multiplication. If we let

 

then

 .

Ideals in quadratic fields

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Let   be an ideal in the ring of integers  ; that is, let   be a nonempty subset of   such that for any   and any  ,  . (An ideal   as defined here is sometimes referred to as an integral ideal to distinguish from fractional ideal to be defined below.) If   is an ideal in   then one can find   such any element in   can be uniquely represented in the form   with  . Such a pair of elements in   is called a basis of the ideal  . This is indicated by writing  . The norm of   is defined as

 .

The norm is independent of the choice of the basis.

Some special ideals

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  • The product of two ideals   and  , denoted by  , is the ideal generated by the  -linear combinations of  .
  • A fractional ideal is a subset   of the quadratic field   for which the following two properties hold:
  1. For any   and for any  ,  .
  2. There exists a fixed algebraic integer   such that for every  ,  .
  • An ideal   is called a principal ideal if there exists an algebraic integer   such that  . This principal ideal is denoted by  .

There is this important result: "Given any ideal (integral or fractional)  , there exists an integral ideal   such that the product ideal   is a principal ideal."

An equivalence relation in the set of ideals

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Two (integral or fractional) ideals   and   ares said to be equivalent, dented  , if there is a principal ideal   such that  . These ideals are narrowly equivalent if the norm of   is positive. The relation, in the set of ideals, of being equivalent or narrowly equivalent as defined here is indeed an equivalence relation.

The equivalence classes (respectively, narrow equivalence classes) of fractional ideals of a ring of quadratic algebraic integers   form an abelian group under multiplication of ideals. The identity of the group is the class of all principal ideals (respectively, the class of all principal ideals   with  ). The groups of classes of ideals and of narrow classes of ideals are called the class group and the narrow class group of the  .

Binary quadratic forms and classes of ideals

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The main result that connects the IBQFs and classes of ideals can now be stated as follows:

"The group of classes of binary quadratic forms of discriminant   is isomorphic to the narrow class group of the quadratic number field  ."

Bhargava's approach to the composition law

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Bhargava cube with the integers a, b, c, d, e, f, g, h at the corners

Manjul Bhargava, a Canadian-American Fields Medal winning mathematician introduced a configuration, called a Bhargava cube, of eight integers   (see figure) to study the composition laws of binary quadratic forms and other such forms. Defining matrices associated with the opposite faces of this cube as given below

 ,

Bhargava constructed three IBQFs as follows:

 

Bhargava established the following result connecting a Bhargava cube with the Gauss composition law:[5]

"If a cube A gives rise to three primitive binary quadratic forms Q1, Q2, Q3, then Q1, Q2, Q3 have the same discriminant, and the product of these three forms is the identity in the group defined by Gauss composition. Conversely, if Q1, Q2, Q3 are any three primitive binary quadratic forms of the same discriminant whose product is the identity under Gauss composition, then there exists a cube A yielding Q1, Q2, Q3."

References

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  1. ^ Carl Friedrich Gauss (English translation by Arthur A. Clarke) (1965). Disquisitiones Arithmeticae. Yale University Press. ISBN 978-0300094732.
  2. ^ D. Shanks (1989). Number theory and applications, volume 265 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Dordrecht: Kluwer Acad. Publ. pp. 163–178, 179–204.
  3. ^ Duncan A. Buell (1989). Binary Quadratic Forms: Classical Theory and Modern Computations. New York: Springer-Verlag. pp. 62–63. ISBN 978-1-4612-8870-1.
  4. ^ Duncan A. Buell (1989). Binary Quadratic Forms: Classical Theory and Modern Computations. New York: Springer-Verlag. ISBN 978-1-4612-8870-1.
  5. ^ Manjul Bhargava (2006). Higher composition laws and applications, in Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006. European Mathematical Society.