A common denominator for the roots of a split polynomial

Let $a/q$ (lowest terms) be one root and put $g(Y)=q^{\deg f}f(Y/q)$, a monic polynomial with integer coefficients. Then $q^{\deg f}\gamma=-g(a)$ is an integer, so $g+q^{\deg f}\gamma$ is monic in $\Z[Y]$, and its rational roots, which are $q$ times the roots of $f+\gamma$, are integers. So every root has denominator dividing $q$; running the argument from each root, the denominators all coincide.