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Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a Polynomial (respectively, Rational Function) in the Elementary Symmetric Functions on those variables.
See also Elementary Symmetric Function
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 2, 1959.
Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., 1968.