I'm stuck dealing with the proof of the existence of tamely ramified extension. Here is the theorem:
Let $K$ be a field complete with respect to a discrete absolute value, and let $E/K$ be a totally ramified extension of degree $n=e(E/K)$. Let the characteristic of the residue field of $K$ be $p>0$ and suppose that $n=n_{0}p^{l}$ with $(n_{0},p)=1$. Then there exists a unique extension $V$ of $K$ with $K\subset V\subset E$ such that $[V:K]=n_{0}$. Moreover, $V=K(\pi^{1/n_{0}})$ where $\pi$ is an element of $K$ such that $\mathfrak{m}_{K}=\pi\mathcal{O}_{K}$.
Summarizing, the proof relies on the polynomial $g(X)=X^{n_{0}}-\pi u$ with $u\in K$ and $|u|=1$.
Here is the argument of the autor: Let $\Pi^{\prime}$ a roof of $g$, and put $V=K(\Pi^{\prime})$; clearly $[V:K]\leq n_{0}$ but $e(V/K)\geq n_{0}$ (since $|\Pi^{\prime}|^{n_{0}}=|\pi|$).
I don't understand why $|\Pi^{\prime}|^{n_{0}}=|\pi|$ implies $e(V/K)\leq n_{0}$ (e(V/K) is the ramification index of $V$ over $K$). I want to say that, since that happens then we must have that $e$ divides $n_{0}$, but in general I think this is false (In my intuition $n_{0}$ could be a multiple of $e(V/K)$). Also, we are working with non normalized valuation, otherwise I woul have $v(\pi)=1$ and $g$ woul be Eisenstein and the result follows.
I have a result that says the following: Let $\alpha$ belong to the separable closure of the ultrametric field K, and suppose that $v(\alpha)=a/n$ with $a$ relatively prime to $n$. Then $K(\alpha)$ has ramification index divisible by $n$ (and, thus $n$ divides [$K(\alpha):K)$. But I don't have the hypothesis.
I tried to use the division algorithm but I couldn't get a contradiction.
Also, I want to use the fact that if the Newton polygon of a polynomial has only one side without any point on it, other than the vertex, then the polynomial is irreducible. The points between $0$ and $n_{0}$ correspond to infinity points, and $v(\pi u)=v(\pi)>0$ which is fixed, so I can use the above result.
I'm asking about author's argument, is it correct? Any hint to see why this is true?
Thanks