On page 85 of Paul Cohn's Algebraic Numbers and Algebraic Functions, Cohn is showing that the minimal polynomial of an algebraic number has integer coefficients.
He says, let $\alpha$ have minimal polynomial $g$ and suppose $\alpha$ satisfies $f=0$, where $f$ is monic with integer coefficients. Since $g$ is the minimal polynomial for $\alpha$, $f=gh$ for some $h$. Choose integers $a$ and $b$ such that $ag$, $bh$ are primitive with integer coefficients...and so on.
My question: How exactly does one know that such $a$ and $b$ exist? I think you could take $a$ and $b$ to be the least common multiples of all the denominators in $g$ and $h$, respectively, so $ag$ and $bh$ would have integer coefficients, but I don't feel sure they are primitive.