  Let $F$ be be an arbitrary field and let $h(n)$ be the Heisenberg algebra of
dimension $2n+1$ over $F$. It was shown by Burde that if $F$ has characteristic
0 then the minimum dimension of a faithful $h(n)$-module is $n+2$. We show here
that his result remains valid in prime characteristic $p$, as long as
$(p,n)\neq (2,1)$.
  We construct, as well, various families of faithful irreducible
$h(n)$-modules if $F$ has prime characteristic, and classify these when $F$ is
algebraically closed. Applications to matrix theory are given.
