Definition:Symmetric Polynomial

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Definition

Let $K$ be a field.

Let $K \left[{X_1, \ldots, X_n}\right]$ be the ring of polynomial forms over $K$.


A polynomial $f \in K \left[{X_1, \ldots, X_n}\right]$ is symmetric if for every permutation $\pi$ of $\left\{{1, 2, \ldots, n}\right\}$:

$f \left({X_1, \ldots, X_n}\right) = f \left({X_{\pi\left({1}\right)}, \ldots, X_{\pi \left({n}\right)}}\right)$

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