# Definition:Symmetric Polynomial

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 $\left\{{1, 2, \ldots, n}\right\}$:
$f \left({X_1, \ldots, X_n}\right) = f \left({X_{\pi\left({1}\right)}, \ldots, X_{\pi \left({n}\right)}}\right)$