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I'm learning about local fields and I'm confused about some basic issues - help gratefully received!


Let $K/k$ be a finite Galois extension of number fields with rings of integers $\mathfrak{O}$ and $\mathfrak{o}$ respectively.

Let $\mathfrak{p}$ be a prime ideal of $\mathfrak{o}$ and let $\mathfrak{P}$ be a prime ideal of $\mathfrak{O}$ lying over $\mathfrak{p}.$

We can consider the extension of complete fields $K_{\mathfrak{P}}/k_{\mathfrak{p}}$ with valuation rings $\mathfrak{O}_{\mathfrak{P}}$ and $\mathfrak{o}_{\mathfrak{p}}$ resp.

Some questions:

1) Is it necessarily true that $K_{\mathfrak{P}}/k_{\mathfrak{p}}$ is Galois?

2) If so, can we compare Gal$(K_{\mathfrak{P}}/k_{\mathfrak{p}})$ and Gal$(L/K)$ in some way?

Some thoughts:

1) I know that this extension is finite of degree at most $[K:k]$ but that's about it...

2) I know that the residue field of $K_{\mathfrak{P}}$ (resp. $k_\mathfrak{p}$) is isomorphic to $\mathfrak{O}/\mathfrak{P}$ (resp. $\mathfrak{o}/\mathfrak{p}$) and so the extension of residue fields is Galois with group isomorphic to $D_\mathfrak{P}/I_\mathfrak{P}$ where $D_\mathfrak{P}$ (resp. $I_\mathfrak{P}$) is the decomposition group (resp. inertia group) of $\mathfrak{P}$ in $K/k.$ But this might be irrelevant to the question I ask - I'm not sure...

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  • $\begingroup$ The buzzword is "decomposition group". $\endgroup$ – Lord Shark the Unknown Oct 13 '17 at 14:34
  • $\begingroup$ I mentioned that in my 2nd point in "some thoughts" but I'm not sure how it relates to my question - could you give a reference? $\endgroup$ – user350031 Oct 13 '17 at 14:34
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The answers to your questions can be summarized in a concise way as e.g. in Cassels-Fröhlich, chap. VII by Tate :

Let $K/k$ be a finite Galois extension of number fields (more generally, of global fields). A prime of $K$ (resp. of $k$) is an equivalence class of valuations (archimedean or not) on $K$ (resp. on $k$), denoted $w$ (resp. $v$). If the prime is not archimedean, it comes from a $\mathfrak p$-adic valuation as in your questions. The group $G =Gal(K/k)$ permutes naturally the primes of $K$, and any automorphism $s \in G $ induces by continuity an isomorphism $s_w : K_w \cong K_{sw}$. If $w$ of $K$ lies above $v$ of $k$, so does $sw$ and $s_w$ is actually a $K_v$-isomorphism. Define the decomposition subgroup $G_w$ of $G$ to be the isotropy subgroup of $w$, i.e. $G_w$={$s\in G \mid sw=w$}. Although $G_w$ is determined by $w$ only up to conjugacy (i.e. $G_{tw}=tG_wt^{-1}$), any $s \in G_w$ is a $k_v$-automorphism of $K_w$ for $w$ above $v$, so that $G_w$ injects into $Gal(K_w /k_v)$. Then (op. cit., propos. 1.2):

(i) $G$ permutes transitively the primes $w$ of $K$ over a given prime $v$ of $k$

(ii) $K_w/k_v$ is Galois, and $G_w \cong Gal(K_w/k_v)$ via the above injection

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Yes, $K_{\mathfrak{P}}/k_{\mathfrak{p}}$ is again Galois, and its Galois group is isomorphic to the decomposition group of $\mathfrak{P}$ (over $\mathfrak{p}$). You can find this, for example, in Neukirch's $\textit{Algebraic Number Theory}$ in Chapter II, §$9$. "Galois Theory of Valuations".

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  • $\begingroup$ Many thanks! :) $\endgroup$ – user350031 Oct 13 '17 at 15:02

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