# Symmetric Function

(redirected from*Fundamental theorem of symmetric functions*)

## symmetric function

[sə′me·trik ′fəŋk·shən]## Symmetric Function

a function of two or more variables that remains unchanged under all permutations of the variables. Examples are and 4x_{1}*x*_{2}*x*_{3}

Of particular importance in algebra are symmetric polynomials, especially elementary symmetric polynomials—that is, the functions

where the summations are extended over all combinations of unequal numbers *k*, l, …. The sums are linear in each of the variables. According to the Vieta formulas, *x*_{1}, *x*_{2},…,*x*_{n} are the roots of the equation

*x*^{n} – *f*_{1} x^{n-1} + *f*_{2} x^{n-2} – … + (–1)^{n}*f*_{n} = 0

The fundamental theorem of the theory of symmetric polynomials states that any symmetric polynomial can be represented in one, and only one, way as a polynomial in the elementary symmetric polynomials: *F* (*x*_{1}, *x*_{2},…. *x*_{n}) = *G* (*f*_{1}, *f*_{2},…, *f*_{n}). If all the coefficients of Fare integers, then so are the coefficients of *G*. Thus, every symmetric polynomial with integer coefficients on the roots of an equation can be expressed as a polynomial with integer coefficients on the coefficients of that equation; for example.

Another important class of symmetric functions are the power sums

The relation between such power sums and elementary symmetric polynomials is given by the Newton formulas:

and

These formulas permit the *f*_{k} to be expressed successively in terms of the *s*_{m}, and vice versa.

A function is said to be skew-symmetric, or alternating, if it remains unchanged under even permutations of *x*_{1}, *x*_{2}, …., *x*_{n} and changes sign under odd permutations. Such functions can be rationally expressed in terms of *f*_{1}, *f*_{2},…, *f*_{n} and the product (*see*DISCRIMINANT)

the square of which is a symmetric function and thus can be rationally expressed in terms of *f*_{1}, *f*_{2},…,*f*_{n}

### REFERENCCE

Kurosh, A. G.*Kurs vysshei algebry*, 1 10th ed. Moscow, 1971.