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Let $\widehat{L}/\widehat{K}$ be an extension of local field we know there are a number field $L$ and a place $\frak P$ of $L$ such that $\widehat{L}=L_{\frak P}.$

1) How we can prove that $\widehat{L}=\widehat{K}.L$ (in fact we must show that $L$ dense in $\widehat{K}.L$ ) ?

Denote $G=\mathrm{Gal}(\widehat{L}/\widehat{K})$

2) why $\sigma\longrightarrow\sigma_{|L}$ is an injective homomorphism of $G$ into the automorphism group of $L$?

thank you for your help.

Med
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  • I'm not sure, but I would think that in part 1) you can use the fact that $L$ is a finite dimensional vector space over $K$, isn't it? For part 2) I would first try to prove (if not already known) that any $\sigma$ is continuous. Then use the general principle that a continuous function is determined by its restriction to a dense subset. I'm sure other would appreciate your question more, if you commented on whether there are difficulties in this IMHO natural approach. There are some members, who view your question as a "no-effort shown HW problem." That presses people's buttons here. – Jyrki Lahtonen Nov 17 '13 at 17:00
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    is not a Homework !! – Med Nov 17 '13 at 17:09
  • @JyrkiLahtonen you can give me a detailed answer, and thank you – Med Nov 17 '13 at 17:10
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    I believe you. And the question is sufficiently advanced that I don't care even if it were :-) Other members feel differently though, and it cannot really do any harm trying to take that into account. But: is it given/known that $[L:K]<\infty$? – Jyrki Lahtonen Nov 17 '13 at 17:18

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I'm assuming that $L/K$ is finite dimensional. It's false otherwise. For example if $L$ is the algebraic closure of $K=Q$, then the completion of $L$ is the field usually called $C_p$. But $C_p$ is definitely not $Q_pL$.

The field $L$ has two norms of interest in this situation. The first is the $p$-adic norm coming from the place. The second is the "sup"-norm on $L$ obtained by choosing a basis for $L$ over $K$ and defining $|x|$ to be the maximum of the $K$-norms of the coefficients of $x$ in this basis.

Now you need the theorem which says that any two norms on a finite dimensional vector space over a locally compact field are equivalent (discussed here for example.) From this theorem you conclude that the completion of $L$ in the $p$-adic topology is the same as the completion of $L$ with $|\cdot|$. But the completion with $|\cdot|$ is $\hat{K}L$ pretty much by definition.

For the second one, the point is that the Galois group $Gal(\hat{L}/\hat{K})$ acts through continuous maps, so if $\sigma_L$ is trivial, it means that $\sigma$ fixes $K$; and since $\sigma$ is continuous, it fixes $\hat{K}$, and is thus trivial.

Community
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Jeremy Teitelbaum
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  • thank you Jeremy Teitelbaum , I understand the first part but not the second if you can explain to me a bit more – Med Nov 17 '13 at 20:09
  • The claim is that, if two continuous maps from $X\to Y$ agree on a dense subset of $X$, then they agree everywhere. The action of the Galois group is continuous because, in fact, $|\sigma(x)|=|x|$ -- it acts by isometries. – Jeremy Teitelbaum Nov 17 '13 at 20:13