  In this paper, we will study the M\"obius polynomial, an invariant of ranked
posets that arises in the study of splitting algebras. We will present a
formula for the M\"obius polynomial of the direct product of posets in terms of
the M\"obius polynomials of the factors. We will then use this formula to
calculate Hilbert series and graded trace generating functions associated to
the splitting algebras of the Boolean algebra and the poset of factors of a
natural number n.
