The linear algebra needed to answer this question is spelled out in my paper Dynamical Properties of Quasihyperbolic Toral Automorphisms (Ergodic Th. & Dynam. Sys. 2 (1982), 49-68), in particular Section 2.
The observation that a monic polynomial with integer coefficients all of whose roots lie on the unit circle must be cyclotomic goes back to Kronecker: Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. reine angew. Math. 53 (1857), 173-175. David Boyd proved a generalization of this for polynomials of several variables, which says that a polynomial whose logarithmic Mahler measure vanishes (for one variable this is equivalent to having all roots on the unity circle) must be a product of generalized cyclotomic polynomials (Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables, J. Number Theory 13 (1981), 116-121.
There are some much deeper results along the following lines. Let $E$ be the direct sum of the generalized eigenspaces of $M$ corresponding to eigenvalues on or inside the unit circle, and assume there are no eigenvalues that are roots of unity. Katznelson showed that there is a constant $C>0$ such that if $v$ is a nonzero integer vector then its Euclidean distance to $E$ is greater than $C\|v\|^{-\dim E}$ (Ergodic Automophisms of $T^n$ are Bernoulli, Israel J. Math. 10 (1971), 186-195). This is the key diophantine component in his proof that all ergodic toral automorphisms are measurably isomorphic to Bernoulli shifts. Subsequently this was extended by me and others to show that all ergodic groups automorphisms are Bernoulli.