Consider a $n \times n$ matrix $A$ with entries from $\mathbb Q$. This acts on a module $M$, a $n$-dimensional vector space, where $x$ acts as $A$. This gives rise to an invariant factor decomposition for $M$.
However, now if we consider the same matrix $A$ acting on a $n$-dimensional complex vector space, is the invariant factor decomposition for $M$ still the same? I think so, because the algorithm for computing invariant factor decomposition did not involve factorizing the characteristic polynomials at any step. Can someone please confirm?