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    Fundamental theorem of symmetric polynomials

    P-E7PIY

    Area
    Algebra
    Topics
    Polynomials, Symmetric Functions
    Status
    draft

    State the fundamental theorem of symmetric polynomials.

    Solution

    Let \(R\) be a commutative ring and let \[ e_1,\dots,e_n \] denote the elementary symmetric polynomials in \(R[x_1,\dots,x_n]\).

    The fundamental theorem of symmetric polynomials says that every symmetric polynomial \[ f\in R[x_1,\dots,x_n]^{S_n} \] can be written uniquely in the form \[ f=F(e_1,\dots,e_n) \] for some polynomial \[ F\in R[t_1,\dots,t_n]. \] Equivalently, \[ R[x_1,\dots,x_n]^{S_n}=R[e_1,\dots,e_n], \] and the elementary symmetric polynomials are algebraically independent over \(R\).

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