A symmetric function of is one whose value is unchanged if are permuted arbitrarily. For example, each of the following is a symmetric function of three variables:
Certain symmetric functions serve as building blocks for all the rest. Let
where the sum is taken over all choices of the indices from .
Then is called the th elementary symmetric function of .
Every symmetric polynomial function of is a polynomial function of . (And every rational function of is a rational function of .)
E.g. For the elementary symmetric functions are
and the above examples expressed in terms of these are:
and let be the th elementary symmetric function of the . Then
Explanation. On the LHS of the equation
the coefficient of is ; on the RHS, the coefficient of is times the sum of all products of of the . Thus .
Solution. The LHS of each equation is a symmetric function of , , . This suggests that we can use the information given to construct a polynomial equation whose roots are , , . Let
from which we find
Thus , , are the roots of the cubic equation
Observe that is one of the roots. Now we can factor to obtain
and so find the complete solution set:
where and are the two complex cube roots of .
Lozansky, Rousseau – Winning Solutions (1996)