I think I'm being a bit slow here.
Lemma: Every algebraic integer is the root of some monic irreducible polynomial with coefficients in $\mathbb Z$.
Corollary: The only algebraic integers in $\mathbb Q$ are the ordinary integers.
I'm struggling to see how this corollary follows from the lemma. Suppose $\alpha$ is an algebraic integer. Then there is a monic, irreducible $f$ with integer coefficients such that $f(\alpha) = 0$. Why can't $\alpha$ be a non-integer?