Gauss composition law

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.

are all integers, is called an integral binary quadratic form (IBQF).

is called a fundamental discriminant if and only if one of the following statements holds If

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

is used to denote the fact that the two forms are equivalent.

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

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

{\displaystyle xu,xv,yu,yv}

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

It may be noted that the composite of two IBQFs, if it exists, is not unique.

Consider the following binary quadratic forms: Let We have These six numbers have no common divisors other than ±1.

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

Then the following equation defines a well-defined binary operation "

The following sketch of the modern approach to the composition law of IBQFs is based on a monograph by Duncan A.

[4] The book may be consulted for further details and for proofs of all the statements made hereunder.

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

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

The set of quadratic algebraic integers of radicand

is a ring under ordinary addition and multiplication.

λ α + μ β ∈

is defined as The norm is independent of the choice of the basis.

There is this important result: "Given any ideal (integral or fractional)

These ideals are narrowly equivalent if the norm of

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 main result that connects the IBQFs and classes of ideals can now be stated as follows: Manjul Bhargava, a Canadian-American Fields Medal winning mathematician introduced a configuration, called a Bhargava cube, of eight integers

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]