# Theory of extensions of non-archimedian local fields

I'm searching for a recommendable reference dealing with theory of non-Archimedean local fields where I can find proofs of the following claims about finite extensions $$L/K$$ of non-Archimedean local fields with finite residue fields $$l / k$$. I'm pretty sure that this request might not have a research level but up to now I haven't obtained a satisfying answer asking the same at MSE.

Firstly, we use notations: We denote by $$q \in O_K$$ a uniformizer of $$O_K$$ and $$\pi \in O_L$$ a uniformizer of $$O_L$$. Then $$n= [L:K]=e \cdot f$$ with $$f= [l:k]$$ and $$(q)O_L= (\pi)^eO_L$$.

Now I'm looking for rigorous proofs of the following statements:

(i) Case $$L/K$$ is Galois & unramified ($$\Rightarrow$$ $$e=1$$ & $$[L:K]= [l:k]$$) and $$p$$ the characteristic of finite field $$k$$:

The claim is $$G= \operatorname{Gal}(L/K)= \operatorname{Gal}(l/k)=g$$. I'm looking for a proof making an explicit construction of the "lift" $$\operatorname{Gal}(l/k) \to \operatorname{Gal}(L/K)$$, i.e. if we have a $$k$$-automorphism of $$l= k(a)= k[X]/ (X^f-1)$$, how can it be uniquely lifted to a $$K$$-automorphism of $$L$$?

(ii) If $$L/K$$ is unramified and $$K= \mathbb{Q}_p$$, then $$L$$ is cyclotomic: $$L= \mathbb{Q}_p(\zeta_n)$$, for $$\zeta_n$$ an appropriate root of unity.

(iii)(1) Case $$L/K$$ totally ramified with $$n =e$$ coprime to $$p=\operatorname{char}(k)$$: there exists $$b \in K$$ with $$L= K(\sqrt[e]{b})$$.

How can this $$b$$ be constructed? Can the $$b$$ be chosen as a uniformizer: i.e. $$b= uq$$ with $$u \in O_K^\times$$?

(2) If moreover $$p \mid \lvert k \rvert -1$$, why is $$L/K$$ cyclic & Galois extension?

See Fesenko and Vostokov - Local fields and their extensions. (i) is Proposition 3.3(2). (ii) is Proposition 3.2(1). (iii)(1) is Proposition 3.5(1) (and, yes, $$b$$ may be chosen as a uniformiser). For (iii)(2), I think you meant $$e \mid \lvert k\rvert - 1$$, not $$p \mid \lvert k\rvert - 1$$ (which is impossible). Then ($$k$$, hence) $$K$$ contains an $$e$$th root of unity, so $$L = K(\sqrt[e]q)$$ is (Galois and) cyclic over $$K$$.
• Hi, thank you for your answer. A couple remarks: (you are right, of course, in (iii) I wanted to require $e \mid \lvert k\rvert - 1$). This requirement implies by finiteness of $k$ that $k$ contains an $e$-th root of unity. Why this imply that $K$ has also one? Do you use at this point the Hensel's lemma? Or is it a more more elementary result? Futhermore, you continue, that this imply that $L = K(\sqrt[e]q)$ (<= here we use already (iii)(1) ) is cyclic? How do you deduce this? – Rachmaninow98 May 21 at 17:50
• Indeed, I use Hensel's lemma to deduce that $K$ contains a (I forgot to say primitive) $e$th root of unity because $k$ does. It is a general fact in Galois theory (nothing to do with valued fields) that a radical extension is cyclic if and only if the ground field contains an appropriate root of unity: en.wikipedia.org/wiki/Radical_extension#Solvability_by_radicals . In this case, $X^e - q$ is irreducible by Eisenstein's criterion. – LSpice May 21 at 18:16
• A nitpick: The proof in Proposition 3.3(2) that gives an answer two my (i) is indeed quite understandable (verify $Gal(L/K) \to Gal(l/k)$ surjective + $\vert Gal(L/K) \vert =\vert Gal(l/k) \vert$, but I'm quite curious if it also possible to build expliitely a lift $\operatorname{Gal}(l/k) \to \operatorname{Gal}(L/K)$, since I want to understand what this lift does "concretly" on $L$. Unfortunately, the proof 3.3(2) from the book not gives this "inside look". Do you know if it also possible to argue via these "lifts"? I mean every $\alpha \in Gal(l/k)$ is a power of – Rachmaninow98 May 21 at 20:45
• Frobenius $a \mapsto a^d$ with $d= \vert k \vert$. That is $\operatorname{Gal}(L/K)$ has also to be generated by a lift of this Frobenius. But I have no idea how this lift of the Frobenius in $Gal(L/K)$ shold act. My first idea was to do Teichmüller lifts of $\zeta_{d-1} \in l$ to $L$ and define to lift of the Frobenius on only on these Teichmüller lifts. Problem: do the Teichmüller lifts generate $L$ as $K$-algebra? Do you probably see another possibly more conventional approach to lift the Frobenius and thus to obtain the desired map $\operatorname{Gal}(l/k) \to \operatorname{Gal}(L/K)$? – Rachmaninow98 May 21 at 20:45