Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds

We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of construct ing matrices in GL(n; Z) whose eigenvalues satisfy certain conditions. Afte rwards, we show how this translation can then be solved in some specific si tuations. The paper starts with a section on polynomial permutations of Q(K ), a subject which is of interest on its own.