A well-known result by Liouville shows that some elementary functions of a real or complex variable does not admit elementary antiderivatives, the usual example being $\exp(-x^2)$. In 1968, Risch gave an algorithm to decide if an elementary function admit an elementary antiderivative. Elementary functions can be defined in term of differentials fields which allow a natural definition over fields of finite characteristic. We discuss the problem of elementary antiderivative in this context, and we extent it to elementary solutions to linear differential equations.