Valuation of $\zeta-1$

Question

Let $p$ be a prime number and let $\mathbb{Q}_p$ be the field of p-adic integer. Let $\zeta$ be a primitive $q$-th root of unity , I know that there is a unique valutation on $\mathbb{Q}_p(\zeta)$ such that this valuation restrict to $\mathbb{Q}_p$ is the valuation $\nu_p$. I want to calcolate the valuation of $\zeta -1 $. I think this is zero but I don’t know how to prove it. Can anyone help me?

Answer 1

Check that $$f(x)=\sum_{n=0}^{p-1} (x+1)^{p^{k-1} n}$$ is Eisenstein at $p$,

from which $f$ is irreducible over $\Bbb{Q}_p$ and the valuation of its root $\zeta_{p^k}-1$ is $$v(\zeta_{p^k}-1)=\frac{v(f(0)}{\deg f} = \frac1{(p-1)p^{k-1}}$$