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

Famous quotes containing the words fundamental and/or theorem:

    This declared indifference, but as I must think, covert real zeal for the spread of slavery, I can not but hate. I hate it because of the monstrous injustice of slavery itself. I hate it because it deprives our republican example of its just influence in the world ... and especially because it forces so many really good men amongst ourselves into an open war with the very fundamental principles of civil liberty.
    Abraham Lincoln (1809–1865)

    To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.
    Albert Camus (1913–1960)