# Unramified extension of $K_π$ and $K_{π,n}$

I have some questions while reading Milne’s CFT.

the fields below are assumed to be separable.

Here $$K$$ is a local field,$$π$$ is a fixed prime element of $$K$$ and $$K_π$$ is the sub-field of $$K^{ab}$$ fixed by $$π$$ who acts on $$K^{ab}$$ by the local Artin map.Also In Milne’s CFT he constructs the $$K_{π}$$ explicitly as a union of $$K_{π,n}$$’s which are totally unramified extension extensions over $$K$$ and of finite degree.

For example,take $$K=Q_p,π=p$$,then $$K_π=(Q_p)_p=\cup_n Q_p[ζ_{p^n}]$$ where $$ζ_m$$ is a primitive m-th root and $$K_{π,n}=Q_p[ζ_{p^n}]$$.

Here are my questions:

1.suppose $$K_π\subseteq L$$ is a finite unramified extension(a priori contained in $$K^{ab}$$),why there exists a finite unramified extension $$K_{π,n}\subseteq L’$$ with $$L=K_π L’$$?

2.suppose $$K_{π,n}\subseteq L’$$ is a finite unramified extension,why there exists a finite unramified extension $$K\subseteq L’’$$ with $$L’=K_{π,n}L’’$$?

The facts are used in lemma 4.10 and I don’t know how to prove.Here is what I have done:

3.for 1,we can assume $$L=K_π[α]$$ for some $$α$$ with the minimal polynomial $$f(x)\in K_π[x]$$.Since $$K_π$$ is the union of $$K_{π,n}$$,for some $$n$$, $$f(x)\in K_{π,n}[x]$$ and then set $$L’=K_{π,n}[α]$$.Obviously $$K_πL’=L$$,but I don’t see why $$L’/K_{π,n}$$ is unramified.

4.for 2, set $$L’’$$ as the largest unramified extension of $$K$$ in $$L’$$, then by construction $$L’’$$ is unramified but again I don’t see why $$K_{π,n}L’’=L’$$.

So are the $$L’, L’’$$ given in 3. and 4. correct? How to prove 1. and 2.? Any help will be appreciated.