# Definition:Elementary Symmetric Polynomial

Let $K$ be a field.
Let $K \left[{X_1, \ldots, X_n}\right]$ be the ring of polynomial forms over $K$.
The elementary symmetric polynomials in $n$ variables are:
$\displaystyle f_r \left({X_1, \ldots, X_n}\right) = \sum_{1 \mathop \le i_1 \mathop < \cdots \mathop < i_r \mathop \le n} x_{i_1} \cdots x_{i_r}: \quad r = 1, \ldots, n$