  We study the structure of the category of graded, connected,
countable-dimensional, commutative and cocommutative Hopf algebras over a
perfect field $k$ of characteristic $p$. Every $p$-torsion object in this
category is uniquely a direct sum of explicitly given indecomposables. This
gives rise to a similar classification of not necessarily $p$-torsion objects
that are either free as commutative algebras or cofree as cocommutative
coalgebras. We also completely classify those objects that are indecomposable
modulo $p$.
