  We show that the infinite symmetric product of a connected graded-commutative
algebra over the rationals is naturally isomorphic to the free
graded-commutative algebra on the positive degree subspace of the original
algebra. In particular, the infinite symmetric product of a connected
commutative (in the usual sense) graded algebra over the rationals is a
polynomial algebra. Applied to topology, we obtain a quick proof of the
Dold-Thom theorem in rational homotopy theory for connected spaces of finite
type. We also show that finite symmetric products of certain simple free graded
commutative algebras are free; this allows us to determine minimal Sullivan
models for finite symmetric products of complex projective spaces.
