2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/213290We 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.8 pages; to appear in the Proc. Amer. Math. SocRings and Algebras16W30text