  Let F be a non-trivial finite extension of the p-adic numbers, and G be a
compact p-adic Lie group whose Lie algebra is isomorphic to a split simple
F-Lie algebra. We prove that the mod p Iwasawa algebra of G has no modules of
canonical dimension one. One consequence is a new upper bound on the Krull
dimension of the Iwasawa algebra. We also prove a canonical dimension-theoretic
criterion for a mod p smooth admissible representation to be of finite length.
Combining our results shows that any smooth admissible representation of
$GL_n(F)$, with central character, has finite length if its dual has canonical
dimension two.
