The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring Ax] in the variable 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 Ax] such that A _k kx]_(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 kx]_(x).