  We introduce symmetrizing operators of the polynomial ring $A[x]$ in the
varible $x$ over a ring $A$. When $A$ is an algebra over a field $k$ these
operators are used to characterize the monic polynomials $F(x)$ of degree $n$
in $A[x]$ such that $A\otimes_k k[x]_{(x)}/(F(x))$ is a free $A$-module of rank
$n$. We use the characterization to determine the Hilbert scheme parameterizing
subschemes of length $n$ of $k[x]_{(x)}$.
