  Let $K$ be an {\em arbitrary} field of characteristic $p>0$ and $\CD (P_n)$
be the ring of differential operators on a polynomial algebra $P_n$ in $n$
variables. A long anticipated {\em analogue of the inequality of Bernstein} is
proved for the ring $\CD (P_n)$. On the way, analogues of the concepts of
  (Gelfand-Kirillov) {\em dimension, multiplicity, holonomic modules} are found
in prime characteristic (giving answers to old questions of finding such
analogs).An analogue of the {\em Quillen's Lemma} is proved for simple {\em
finitely presented} $\CD (P_n)$-modules. In contrast to the characteristic zero
case where the Geland-Kirillov dimension of a nonzero finitely generated $\CD
(P_n)$-module $M$ can be {\em any natural} number from the interval $[n,2n]$,
in the prime characteristic, the
  (new) dimension $\Dim (M)$ can be \underline{{\em any real}} number from the
interval $[n,2n]$.
