Symmetric Functions over Finite Fields

TL;DR

This paper investigates algebraic relations among elementary symmetric polynomials over finite fields, providing a method to find all such relations and proving that the basis polynomials have coefficients in the prime field.

Contribution

It introduces an algorithm to compute all algebraic relations among elementary symmetric polynomials over finite fields and proves the coefficients of the basis polynomials lie in the prime field.

Findings

Computed the number of independent algebraic relations

Developed an algorithm to find all relations

Proved basis polynomials have coefficients in F_p

Abstract

The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of algebraic relations found by the algorithm consists of polynomials having coefficients in the prime field F_p.