  For a general ordinary differential operator $\mathcal{L}$ with periodic
coefficients we prove that the characteristic polynomial of the Floquet matrix
is irreducible over the field of meromorphic functions. We also consider a
multipoint eigenvalue problem and show that its eigenspaces are spanned by pure
or generalized Floquet solutions. Finally, at the end of the paper we mention
some relevant conjectures and open questions.
