 Solution

Suppose that $p$ divided the finite index $[{\mathcal{O}}_K:\mathbb{Z}[\alpha ]]$. Then the quotient would contain nonzero $p$-torsion, so there would be $\beta \in {\mathcal{O}}_K\setminus \mathbb{Z}[\alpha ]$ withwhere not all $b_j$ are divisible by $p$. Let $k$ be the least index for which $p\nmid b_k$. Since $v_P(\alpha )=1$ and $v_P(p)=n$, the term $b_k{\alpha }^k$ is the unique term on the right of least $P$-valuation. ThusBut $\beta$ is an algebraic integer, so $v_P(\beta )\ge 0$ anda contradiction. Therefore