Algebraic Invariants for Group of Permutations of Variables

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Theorem

Let $S = \set {x_1, x_2, \ldots,x_n}$ be a set of algebraic variables.

The algebraic invariants for the group of permutations of $S$ are those generated by the elementary symmetric polynomials:

\(\ds p_1\) \(=\) \(\ds x_1 + x_2 + \cdots + x_n\)
\(\ds p_2\) \(=\) \(\ds x_1 x_2 + x_1 x_3 + \cdots + x_{n - 1} x_n\)
\(\ds \) \(\vdots\) \(\ds \)
\(\ds p_n\) \(=\) \(\ds x_1 x_2 \cdots x_n\)


Proof


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Sources

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