  Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete
local fields with perfect residue fields of characteristic $p>0$. In this
paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$
(abelian or not) and for $K$ of either possible characteristic (0 or $p$), that
are sufficient for the existence of a Galois scaffold. The existence of a
Galois scaffold makes it possible to address questions of integral Galois
module structure, which is done in a separate paper. But since our conditions
can be difficult to check, we specialize to elementary abelian extensions and
extend the main result of [G.G. Elder, Proc. A.M.S. 137 (2009), 1193-1203] from
characteristic $p$ to characteristic 0. This result is then applied, using a
result of Bondarko, to the construction of new Hopf orders over the valuation
ring $\mathfrak{O}_K$ that lie in $K[G]$ for $G$ an elementary abelian
$p$-group.
