Algebraic identities

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Fibonacci identity

Since(1)(2)it follows that(3)(4)This identity implies the two-dimensional Cauchy'sinequality.

Lagrange's identity

Lagrange's identity is the algebraic identity(1)(Mitrinović 1970, p. 41; Marsden and Tromba 1981, p. 57; Gradshteyn and Ryzhik 2000, p. 1049).Lagrange's identity is a special case of the Binet-Cauchy identity, and Cauchy's inequality in dimensions follows from it.It can be coded in the Wolfram Languageas follow. LagrangesIdentity[n_] := Module[ {aa = Array[a, n], bb = Array[b, n]}, Total[(aa^2) Plus @@ (bb^2)] == Total[(a[#1]b[#2] - a[#2]b[#1])^2& @@@ Subsets[Range[n], {2}]] + (^2 ]Plugging in gives the and identities (2)(3)(4)A vector quadruple product formula knownas Lagrange's identity given by(5)(Bronshtein and Semendyayev 2004, p. 185).A related identity also known as Lagrange's identity is given by defining and to be -dimensional vectors for , ..., . Then(6)(Greub 1978, p. 155), where denotes a cross product, denotes a dot product, and is the determinant of the matrix..

Completing the square

The conversion of a quadratic polynomial of the form to the formwhich, defining and , simplifies to

Brioschi quintic form

Using a Tschirnhausen transformation, the principal quintic form can be transformed to the one-parameter form(1)named after Francesco Brioschi (1824-1897) and which is important to the Klein's solution of the general quintic in terms of hypergeometric functions (Doyle and McMullen). This can be attained by using the transformation,(2)(Dickson 1959) and eliminating the variable between the two using resultants to form a new quintic(3)where(4)(5)(6)Equating coefficients with a generic principal quintic(7)results in a system of three equations in the three unknowns , , and . Amazingly, this can be resolved to a single equation that is only a quadratic and given in the variable by(8)(Dickson 1959).

Gauss's polynomial identity

For even ,(1)(Nagell 1951, p. 176). Writing out symbolically,(2)which gives(3)where is a q-Pochhammer symbol.For example, for ,(4)and for ,(5)

Gauss's cyclotomic formula

Let be a prime number, thenwhere and are homogeneous polynomials in and with integer coefficients. Gauss (1965, p. 467) gives the coefficients of and up to .Kraitchik (1924) generalized Gauss's formula to odd squarefree integers . Then Gauss's formula can be written in the slightly simpler formwhere and have integer coefficients and are of degree and , respectively, with the totient function and a cyclotomic polynomial. In addition, is symmetric if is even; otherwise it is antisymmetric. is symmetric in most cases, but it antisymmetric if is of the form (Riesel 1994, p. 436). The following table gives the first few and s (Riesel 1994, pp. 436-442).51711

Bhargava's theorem

Let the th composition of a function be denoted , such that and . Denote the composition of and by , and define(1)Let(2)(3)(4)and(5)(6)Then if (i.e., ),(7)(8)where and composition is done in terms of components.

Multiplicative identity

In a set equipped with a binary operation called a product, the multiplicative identity is an element such thatfor all . It can be, for example, the identity element of a multiplicative group or the unit of a unit ring. In both cases it is usually denoted 1. The number 1 is, in fact, the multiplicative identity of the ring of integers and of its extension rings such as the ring of Gaussian integers , the field of rational numbers , the field of real numbers , and the field of complex numbers . The residue class of number 1 is the multiplicative identity of the quotient ring of for all integers .If is a commutative unit ring, the constant polynomial 1 is the multiplicative identity of every polynomial ring .In a Boolean algebra, if the operation is considered as a product, the multiplicative identity is the universal bound . In the power set of a set , this is the total set .The unique element of a trivial ring is simultaneously the additive identity and multiplicative..

Ford's theorem

Let , , and be integers with . For , 1, 2, letThen

Algebraic identity

An algebraic identity is a mathematical identity involving algebraic functions. Examples include the Euler four-square identity, Fibonacci identity, Lebesgue identity, and the curious identitydue to Y. Kohmoto.

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