  We construct the Weil functor $T^A$ corresponding to a general Weil algebra
$A = K \oplus N$: this is a functor from the category of manifolds over a
general topological base field or ring $K$ (of arbitrary characteristic) to the
category of manifolds over $A$. This result simultaneously generalizes results
known for ordinary, real manifolds, and previous results by the first author
for the case of the higher order tangent functors ($A = T^k K$) and for the
case of jet rings ($A = K[X]/(X^{k+1})$). We investigate some algebraic aspects
of these general Weil functors ("K-theory of Weil functors", action of the
"Galois group" $\Aut_K(A)$), which will be of importance for subsequent
applications to general differential geometry.
