We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in
G
L
(
n
,
Z
)
\mathrm {GL}(n,\mathbb {Z})
whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of
Q
K
\mathbb {Q}^K
, a subject which is of interest on its own.