  It is proved that, if $K$ is a complete discrete valuation field of mixed
characteristic $(0,p)$ with residue field satisfying a mild condition, then any
abelian variety over $K$ with potentially good reduction has finite
$K(K^{1/p^\infty})$-rational torsion subgroup. This can be used to remove
certain conditions assumed in some theorems in Iwasawa theory.
