A common denominator for the roots of a split polynomial

Let $f\in\Z[X]$ be monic and $\gamma\in\Q$ . Show that if $f+\gamma$ splits over $\Q$ , then all its roots have the same denominator in lowest terms.

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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.

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