  The theory of differential equations has an arithmetic analogue in which
derivatives are replaced by Fermat quotients. One can then ask what is the
arithmetic analogue of a linear differential equation. The study of usual
linear differential equations is the same as the study of the differential
cocycle from $GL_n$ into its Lie algebra given by the logarithmic derivative.
However we prove here that there are no such cocycles in the context of
arithmetic differential equations. In sequels of this paper we will remedy the
situation by introducing arithmetic analogues of Lie algebras and a skew
version of differential cocycles; this will lead to a theory of linear
arithmetic differential equations.

Number Theory