Definition:Symmetric Polynomial

Definition

Let $K$ be a field.

Let $K \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomial forms over $K$.

A polynomial $f \in K \sqbrk {X_1, \ldots, X_n}$ is symmetric if and only if for every permutation $\pi$ of $\set {1, 2, \ldots, n}$:

$\map f {X_1, \dotsc, X_n} = \map f {X_{\map \pi 1}, \dotsc, X_{\map \pi n} }$

Also see

• Definition:Elementary Symmetric Polynomial

• Results about symmetric polynomials can be found here.