Question about S.Lang's proof of Kummer's Lemma

Question

I have a question about the proof of Kummer's Lemma in Serge Lang's Cyclotomic fields (i.e. Theorem 6.1). Let $K = \mathbf{Q}(\xi_p)$ the $p$-th cyclotomic field extension of $\mathbf{Q}$. Let $u$ be a unit in $K$. He first shows that it is unramified at the prime ideal $(\xi_p - 1)$ of $K$ .Then he states that $K(u^{1 / p})$ is "trivially unramified" at all other primes of $K$. Why is that?

Thanks!

Answer 1

First, there's a typo in the question; I think you mean $K(u^{1/p})/K$ is unramified at the ideal over $(\zeta_p - 1)$.

To answer the question, the minimal polynomial of this extension certainly divides $x^p - u$, but that polynomial is separable mod $\lambda$ for any $\lambda \nmid p$.

Answer 2

Primes ramified in ${\mathbb Q}(\sqrt[n]{m})$ divide $mn$ by a simple computation of the discriminant $x^n - m$. Now lift the proof to $K$.