  How to extend Beurling's theorem on the shift invariant subspaces of Hardy
class $H^2$ of the unit disk to several complex variables has been an open
problem at least since 1964. In this paper, we prove a generalization of
Beurling's theorem to valuation Hilbert modules over valuation algebras. We
shall apply our abstract Beurling's theorem to obtain a complete description of
the closed invariant subspaces of $H^2$ of the polydisk. As we shall show in a
subsequent paper, further consequences of our abstract Beurling's theorem are
complete extensions of Beurling's theorem to many more Hilbert spaces of
analytic functions in several complex variables.
