  In this paper, we prove that the ring of polynomial invariants of the Weyl
group for an indecomposable and indefinite Kac-Moody Lie algebra is generated
by invariant symmetric bilinear form or is trivial depending on $A$ is
symmetrizable or not. The result was conjectured by Moody and assumed by Kac.
As applications we discuss the rational homotopy types of Kac-Moody groups and
their flag manifolds.
