  Using vertex operators, we construct explicitly Lusztig's $\mathbb Z[q,
q^{-1}]$-lattice for the level one irreducible representations of quantum
affine algebras of ADE type. We then realize the level one irreducible modules
at roots of unity and show that the q-dimension is still given by the Weyl-Kac
character formula. As a consequence we also answer the corresponding question
of realizing the affine Kac-Moody Lie algebras of simply laced type at level
one in finite characteristic.
