  The problem of the existence of non-medial distributive hamiltonian
quasigroups is solved. Translating this problem first to commutative Moufang
loops with operators, then to ternary algebras and, finally, to cocyclic
modules over\linebreak $\Bbb Z[x,x^{-1},(1-x)^{-1}]$, it is shown that every
non-medial distributive hamiltonian quasigroup has at least 729 elements and
that there are just two isomorphism classes of such quasigroups of the least
cardinality. The quasigroups representing these two classes are
anti-isomorphic.
