---
type: "concept"
title: "Symmetric polynomial"
slug: "symmetric-polynomial"
language: "en"
translationGroupId: "cmrc5itp2000bs701vwsdo4zv"
domain: "Algebra"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

A symmetric polynomial is a [[polynomial|polynomial]] $P\in k[X_{1},\ldots,X_{n}]$ such that, for every [[permutation|permutation]] $\sigma \in S_{n}$:
$$P(X_{\sigma(1)},\ldots,X_{\sigma(n)})=P(X_{1},\ldots,X_{n}).$$