  Let $G$ be a finite $p$-group and $k$ a field of characteristic $p>0$. We
show that $G$ has a \emph{non-linear} faithful action on a polynomial ring $U$
of dimension $n=\mathrm{log}_p(|G|)$ such that the invariant ring $U^G$ is also
polynomial. This contrasts with the case of \emph{linear and graded} group
actions with polynomial rings of invariants, where the classical theorem of
Chevalley-Shephard-Todd and Serre requires $G$ to be generated by
pseudo-reflections.
  Our result is part of a general theory of "trace surjective $G$-algebras",
which, in the case of $p$-groups, coincide with the Galois ring-extensions in
the sense of \cite{chr}. We consider the \emph{dehomogenized symmetric algebra}
$D_k$, a polynomial ring with non-linear $G$-action, containing $U$ as a
retract and we show that $D_k^G$ is a polynomial ring. Thus $U$ turns out to be
\emph{universal} in the sense that every trace surjective $G$-algebra can be
constructed from $U$ by "forming quotients and extending invariants". As a
consequence we obtain a general structure theorem for Galois-extensions with
given $p$-group as Galois group and any prescribed commutative $k$-algebra $R$
as invariant ring. This is a generalization of the Artin-Schreier-Witt theory
of modular Galois field extensions of degree $p^s$.
