Let $X=\operatorname{Spec} B$ be an affine variety
over a field of arbitrary characteristic, and suppose
that there exists an action of a unipotent group
(possibly neither smooth nor connected). The
fundamental results are as follows.
(1) An algorithm for computing invariants is given,
by means of introducing a degree in the ring of
functions of the variety, relative to the action.
Therefore an algorithmic construction of the quotient,
in a certain open set, is obtained. In the case of a
Galois extension, $k\hookrightarrow B=K$, which is
cyclic of degree $p=\text{ch} (k)$ (that is, such that
the unipotent group is $G={\Bbb Z}/p {\Bbb Z}$),
an element of minimal degree becomes an
Artin--Schreier radical, and the method for computing
invariants gives, in particular, the
expression for any element of $K$ in terms of these
radicals, with an explicit formula. This replaces
the well-known formula of Lagrange (which is valid
only when the degree of the extension and the
characteristic are relatively prime) in the case of
an extension of degree $p=\text{ch}(k)$.
(2) In this paper we give an effective
construction of a stable open subset where there is
a quotient. In this sense we obtain an algebraic local
criterion for the existence of a quotient in a
neighbourhood. It is proved (provided the variety is
normal) that, in the following cases, such an open
set is the greatest one that admits a quotient:
\begin{enumerate}
\item[(a)] when the action is such that the orbits have
dimension less than or equal to 1 (arbitrary
characteristic) and, in particular, for any action
of the additive group $G_a$;
\item[(b)] in characteristic 0, when the action is proper
(obtained from the results of Fauntleroy) or the
group is abelian. 1991 Mathematics Subject Classification:
primary 14L30; secondary 14D25, 14D20.