I'm stuck with the following problem given in a book which I'm reading, it's about creating a tamely ramified extension of $\mathbb{Q}_{p}$.
Let $p\in\mathbb{Z}$ be a prime number, and let $\mathbb{Q}_{p}$ be the field of $p$-adic numbers. Given a finite field extension $K/\mathbb{Q}_{p}$, let $k$ be the residue field of $K$. We define the ramification index $e(K/\mathbb{Q}_{p})$ as the index group of the value group of $K$ in the value group of $\mathbb{Q}_{p}$. Also, we define the inertia/residue degree $f(K/\mathbb{Q}_{p})$ as the degree extension of $k/\mathbb{F}_{p}$. By a theorem, we know that $[K:\mathbb{Q}_{p}]=e(K:\mathbb{Q}_{p})f(K:\mathbb{Q}_{p})$. Finally, we say that $K/\mathbb{Q}_{p}$ is tamely ramified if $p\nmid e(K/\mathbb{Q}_{p})$.
Here is the problem. Consider the polynomial $f=\frac{X^{p}-1}{X-1}=X^{p-1}+X^{p-2}+\cdots+1\in\mathbb{Q}_{p}[X]$. Let $\zeta_{p}$ be a root of $f$, the claim is that $\mathbb{Q}_{p}(\zeta_{p})/\mathbb{Q}_{p}$ is tamely ramified.
Here is my attempt: I have proved that $f$ is irreducible over $\mathbb{Q}_{p}$, so let $\zeta_{p}$ be a root of $f$, then the extension $\mathbb{Q}_{p}(\zeta_{p})/\mathbb{Q}_{p}$ has degree $p-1$. Since $(p,p-1)=1$, in order to prove that the extension is tamely ramified, we need to show thet the residue fiel of $\mathbb{Q}_{p}(\zeta_{p})$ is $\mathbb{F}_{p}$ or that the ramification index of $\mathbb{Q}(\zeta_{p})/\mathbb{Q}_{p}$ agrees with the degree of the extension, i.e., $p-1$.
My attempt is to show that $\mathbb{F}_{p}$ is the residue field of $\mathbb{Q}_{p}(\zeta_{p})$. To do this, let $k$ be the residue field of $\mathbb{Q}_{p}(\zeta_{p})$. It is clear that $\mathbb{F}_{p}\subset k$. So, to see $k\subset\mathbb{F_{p}}$, let $x$ in the valuation ring of $\mathbb{Q}_{p}(\zeta_{p})$, i.e., $|x|\leq 1$. I can write $x=a+b\zeta_{p}$, where $a,b\in\mathbb{Z}_{p}$. If we denote with $\overline{x}$ the class of $x$ in $k$, we have $\overline{x}=\overline{a}+\overline{b}\overline{\zeta_{p}}$. Note that $\overline{a},\overline{b}\in\mathbb{F}_{p}$, so we only need to show $\overline{\zeta_{p}}\in\mathbb{F}_{p}$.
Previously, I proved that $\zeta_{p}=1+\lambda_{1}$, where $|\lambda_{1}|=p^{-1/(p-1)}<1$, hence the class of $\lambda_{1}$ is zero, thus $\overline{\zeta_{p}}=\overline{1}$, which proves that $\overline{x}\in\mathbb{F}_{p}$, so we can conclude that $\mathbb{Q}(\zeta_{p})/\mathbb{Q}_{p}$ is tamely ramified.
Is my argument corect?
I would appreciate any correction or any hint, thanks.