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.


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