Elementary Symmetric Polynomial - The Fundamental Theorem of Symmetric Polynomials

The Fundamental Theorem of Symmetric Polynomials

For any commutative ring A denote the ring of symmetric polynomials in the variables with coefficients in A by .

is a polynomial ring in the n elementary symmetric polynomials for k = 1, ..., n.

(Note that is not among these polynomials; since, it cannot be member of any set of algebraically independent elements.)

This means that every symmetric polynomial P(X_1,\ldots, X_n) \in A^{S_n} has a unique representation

for some polynomial . Another way of saying the same thing is that is isomorphic to the polynomial ring through an isomorphism that sends to for .

Read more about this topic:  Elementary Symmetric Polynomial

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