Stein, Algebraic Number Theory, Proposition 2.4.5

Question

In Stein's proof of propositions 2.4.5 on page 25 of A Computational Approach to Algebraic Number Theory at the end he makes the claim that $\mathcal{O}_K$ has finite index in $$ A = \frac{1}{d}\mathbb{Z}\cdot a_1 + \cdots + \frac{1}{d}\mathbb{Z}\cdot a_n \cong \mathbb{Z}^n $$ and then states that because of this $\mathcal{O}_K$ is free of rank $n$. Why does $\mathcal{O}_K$ not have any torsion?