Timeline for Index of norm $ 1 $ subgroup in a cyclic extension
Current License: CC BY-SA 4.0
17 events
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| Oct 10, 2021 at 12:18 | comment | added | LSpice | That is a different question. As @paulgarrett and @GHfromMO say, you should clarify what you are asking, possibly in a new question. (For example, what kind of characterisation of the image would be acceptable? Can it use global class-field theory?) | |
| Oct 10, 2021 at 5:12 | comment | added | Sky | @Lspice I want to to know the image of $ N(L/\mathbb {Q})$ I.e which points of $ \mathbb{Q}^{\times)$ are in the image set. | |
| Oct 10, 2021 at 1:40 | comment | added | GH from MO | I voted to close, because the index of $U$ in $L^\times$ is infinite ($\omega$) for obvious reasons (as LSpice explained). If the OP meant to ask something else, he should ask it in a separate question. | |
| Oct 9, 2021 at 23:03 | comment | added | paul garrett | @LSpice, I guess some version of the question is just Hilbert's Thm 90 for cyclic extensions... and whether it extends... which it does not, etc. Dunno. | |
| Oct 9, 2021 at 22:46 | comment | added | paul garrett | @LSpice, dunno. My reaction was/is just a charades-sort-of-thing "sounds like" [tugs ear]. There are certainly versions of such a question that have no sensible answers, and some that have trivial... and more interesting ones in-between. Possibly the original asker can refine the question... | |
| Oct 9, 2021 at 22:38 | comment | added | LSpice | @paulgarrett, obviously you have understood this question better than I have. Your comment seems to be about the quotient $\mathbb Q^\times/\operatorname N_{L/\mathbb Q}(L^\times)$, right? But the question seems to be about $L^\times/\ker \operatorname N_{L/\mathbb Q}$—and not even about its structure, but its cardinality, which is infinite, right? What am I missing? | |
| Oct 9, 2021 at 21:19 | comment | added | paul garrett | Well, for cyclic extensions with roots of unity in the basefield, this is Kummer theory. But, more generally, isn't this verging on classfield theory? Already for abelian extensions of local fields, the "norm residue map/theorem" is a serious thing: for unramified extensions it's relatively easy to understand, but not in general. The analogue for global fields (e.g., number fields) is classfield theory! Not an easy or obvious thing. Am I maybe misunderstanding something? (Not all sub-questions of difficult questions are difficult... :) | |
| Oct 9, 2021 at 20:11 | comment | added | LSpice | I'm not sure what map you mean. The norm map $L^\times/U \to \operatorname{Im}(\operatorname N_{L/\mathbb Q})$ is definitely surjective, by the definition of the image, and even an isomorphism; and the image includes $(\mathbb Q^\times)^{2n}$, hence is infinite. I am not claiming that $L^\times/U \to \mathbb Q^\times$ is surjective, and we do not need it to be for my claim. | |
| Oct 9, 2021 at 19:26 | history | edited | LSpice | CC BY-SA 4.0 |
Missing $\times$
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| Oct 9, 2021 at 18:58 | comment | added | Sky | Yes @Lspice if we consider norm map then image of this map will be required quotient group. But does the map subjective? | |
| Oct 9, 2021 at 18:51 | review | Close votes | |||
| Nov 7, 2021 at 22:12 | |||||
| Oct 9, 2021 at 18:49 | history | edited | Sky | CC BY-SA 4.0 |
deleted 15 characters in body
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| Oct 9, 2021 at 18:45 | comment | added | Sky | Yes @Lspice you are right, I have edited my question. | |
| Oct 9, 2021 at 18:43 | history | edited | Sky | CC BY-SA 4.0 |
deleted 15 characters in body
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| Oct 9, 2021 at 18:29 | comment | added | LSpice | Actually $U = \operatorname{Im}(\phi)$. In fact, $L^\times/U$ is naturally isomorphic to $\operatorname{Im}(\operatorname N_{L/\mathbb Q})$, not, as far as I can see, to $\operatorname{Im}(\phi)$. Or did you mean something else? \\ Also, this index will always be infinite in your setting, so are you sure that's what you mean to ask about? | |
| Oct 9, 2021 at 18:28 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Oct 9, 2021 at 18:03 | history | asked | Sky | CC BY-SA 4.0 |