  The Floquet theorem allows to reformulate periodic time-dependent problems
such as the interaction of a many-body system with a laser field in terms of
time-independent, field-dressed states, also known as Floquet states. If this
was possible for density functional theory as well, one could reduce in such
cases time-dependent density functional theory to a time-independent Floquet
density functional theory. We analyze under which conditions the Floquet
theorem is applicable in a density-functional framework. By employing numerical
{\em ab initio} solutions of the interacting time-dependent Schr\"odinger
equation with time-periodic external potentials we show that the exact
effective potential in the corresponding Kohn-Sham equation is {\em not}
unconditionally periodic. Whenever several Floquet states in the interacting
system are involved in a physical process the corresponding
Hartree-exchange-correlation potential is not periodic with the external
frequency only. Using an analytically solvable example we demonstrate that, in
general, the periodicity of the time-dependent Kohn-Sham Hamiltonian cannot be
restored by choosing a different initial state. Only if the external periodic
potential is sufficiently weak such that the initial state of the interacting
system evolves adiabatically to a single, field-dressed state, the resulting
Kohn-Sham system admits the application of the Floquet theorem.
