Timeline for When is the extension $L(S)/L$ Galois and totally ramified?
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24 events
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| Aug 24, 2021 at 16:45 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 21, 2021 at 19:23 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 21, 2021 at 19:22 | comment | added | MAS | @LSpice, sorry, I did another mistake. When I said $\mathcal{O}_L/\pi^n \mathcal{O}_L$, I just mean $n$ is any natural number (not the number of variables). I changed the notation in question | |
| Aug 21, 2021 at 19:14 | comment | added | LSpice | And, although of course a (very) special case is a reasonable starting point, by "some conditions need to be imposed" I meant: how should a conjectural $f$ behave? (Why do you want it? Why do you expect it to exist?) Anyway, if $L(S)/L$ is totally ramified then you have some such map, for some $n$ depending on the 'sauvagerie', but I see no reason $n$ should be the number of variables. | |
| Aug 21, 2021 at 19:10 | comment | added | LSpice | No; I mean that, even if all the obstructions vanish, still you'll get different $\theta$'s for different Galois-group elements. I was writing $\theta_\sigma$ and $\theta_\tau$ for the $\theta$'s associated to $\sigma$ and $\tau$. | |
| Aug 21, 2021 at 18:49 | comment | added | MAS | @LSpice, I just assumed $S$ has $\mathcal{O}_L$-module structure in order to define the map, as you said in your earliar comment to 'impose some conditions'. What do you mean by $\theta_{\tau}$ ? Do you mean $\tau$ acting on $\theta$ i.e., $\tau(\theta)$ ? Finally, Is there any other way to define a homomorphism between the groups ? | |
| Aug 21, 2021 at 18:46 | comment | added | MAS | $\textbf{Correction:}$ In some of my above comments, I meant $\mathcal{O}_L$ not $\mathcal{O}_K$. | |
| Aug 21, 2021 at 18:21 | comment | added | LSpice | It's not clear to me why a $\theta$ as in your definition should exist, nor why it should be independent of $s$; and it seems to me unlikely that $S$ should carry such a module structure. If all these obstructions vanish, then $f$ is clearly a homomorphism, since $\sigma\tau(s) = \sigma(\theta_\tau)\sigma(s) = \theta_\tau\theta_\sigma s$ for all $s \in S$ and all $\sigma, \tau \in \operatorname{Gal}(L(S)/L)$. | |
| Aug 21, 2021 at 17:51 | comment | added | MAS | @LSpice, If I want to define a group homomorphism $f:~Gal(L(S)/L) \to (\mathcal{O}_K/\pi^n\mathcal{O}_K)^{\times}$ which becomes injective. I am not sure how to do it. But I think we need to use the action of $Gal(L(S)/L)$ on $S$ to define the homomorphism. Assume that the zero set $S$ has $\mathcal{O}_L$-module structure. Now, set $f(\sigma)=\theta$ such that $\sigma(s)=\theta * s$ for $s \in S$, $\theta \in \mathcal{O}_L$. Here $*$ is the action of $\mathcal{O}_L$ on the zero set $S$. Does this make $f$ a group homomorphism ? | |
| Aug 21, 2021 at 17:34 | comment | added | LSpice | Well, it depends on what you want; your current question doesn't mention any such map. Of course there is a trivial map $\operatorname{Gal}(L(S)/L) \to (\mathcal O_K/\pi^n\mathcal O_K)^\times$, so probably some conditions need to be imposed. \\ You have also kept the language "$\operatorname{Gal}(L'/L)$ acts transitively on $S$" where you seem just to mean "$\operatorname{Gal}(L'/L)$ acts on $S$". | |
| Aug 21, 2021 at 17:08 | review | Close votes | |||
| Aug 26, 2021 at 3:06 | |||||
| Aug 21, 2021 at 16:58 | comment | added | MAS |
@LSpice, thanks, I have excluded the word proper. But in Chat section, as you said, $L(S)=\bar L$ was in previous setting, slightly different. I am not sure whether in the current setting $L(S)$ is equal to $\bar L$ or not. Second, regarding the level of the question, I want to define a map $Gal(L(S)/L) \to (\mathcal{O}_K/\pi^n \mathcal{O}_K)^{\times}, \ n \geq 1$, once I know $Gal(L(S)/L)$ is Galois. Any help here ?
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| Aug 21, 2021 at 16:53 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 21, 2021 at 16:49 | comment | added | LSpice | The work you do seems mostly fine, aside from some strange terminology (you seem to say that $L(S)$ is a proper subfield of $\bar L$ to mean just that it is a subfield—in fact there's some argument in the chat that it's often all of $\bar L$; and you say that a permutation acts transitively on a set to mean just that it is, well, a permutation of that set), but this does not seem to be a research-level question. | |
| Aug 21, 2021 at 16:48 | history | edited | LSpice | CC BY-SA 4.0 |
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| Aug 21, 2021 at 16:43 | history | edited | MAS | CC BY-SA 4.0 |
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| S Aug 20, 2021 at 19:13 | history | mod moved comments to chat | |||
| S Aug 20, 2021 at 19:13 | comment | added | Stefan Kohl♦ | Comments are not for extended discussion; this conversation has been moved to chat. | |
| Aug 20, 2021 at 17:49 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 20, 2021 at 17:38 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 20, 2021 at 17:09 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 20, 2021 at 12:59 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 20, 2021 at 12:21 | history | edited | MAS | CC BY-SA 4.0 |
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| Aug 20, 2021 at 11:41 | history | asked | MAS | CC BY-SA 4.0 |