Exercise 13, § 5

Let $p$ be a prime number, $f$ an integer $> 0$. Put $L = \mathbf{Q}(\zeta)$ where $\zeta$ is a primitive $p^f$th root of 1, $B = \mathbf{Z}[\zeta]$, the subring of $L$ generated by $\zeta$. Now prove that $B$ is the integral closure of $\mathbf{Z}$ in $L$. (By a discriminant calculation, one will show that $B\left[\frac{1}{p}\right]$ is the integral closure of $\mathbf{Z}\left[\frac{1}{p}\right]$ (V, § 1, No. 6, Lemma 3); denoting by $\overline{B}$ the integral closure of $B$, one will deduce that $\overline{B} \subset \bigcap_q \mathbf{Z}_q[\zeta]$ where $q$ runs through the prime numbers, whence $\overline{B} \subset \mathbf{Z}[\zeta]$.)