  In this article the $p$-essential dimension of generic symbols over fields of
characteristic $p$ is studied. In particular, the $p$-essential dimension of
the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by
$n+\ell$ when the base field is algebraically closed of characteristic $p$. The
proof uses new techniques for working with residues in Milne-Kato
$p$-cohomology and builds on work of Babic and Chernousov in the Witt group in
characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2
symbols) result from this work. The generic $p$-symbol algebra of length $\ell$
is shown to have $p$-essential dimension equal to $\ell+1$ as a $p$-torsion
Brauer class. The second is a lower bound of $\ell+1$ on the $p$-essential
dimension of the functor $\mathrm{Alg}_{p^\ell,p}$. Roughly speaking this says
that you will need at least $\ell+1$ independent parameters to be able to
specify any given algebra of degree $p^{\ell}$ and exponent $p$ over a field of
characteristic $p$ and improves on the previously established lower bound of 3.
