  We prove that a finite-dimensional irreducible Hopf algebra $H$ in positive
characteristic is semisimple, if and only if it is commutative and semisimple,
if and only if the restricted Lie algebra $P(H)$ of the primitives is a torus.
This generalizes Hochschild's theorem on restricted Lie algebras, and also
generalizes Demazure and Gabriel's and Sweedler's results on group schemes, in
the special but essential situation with finiteness assumption added.
