Timeline for Image of norm map for local field
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2011 at 14:01 | comment | added | David Loeffler | Serre's "Local Fields" is the bible here. Milne's notes on local class field theory (from his website) are also very good. As for the subgroups you mention: if t is positive and reasonably large, the p-adic logarithm map will give an isomorphism between $(1 + \pi^t \mathcal{O}_E, \times)$ and $(\pi^t \mathcal{O}_E, +)$, and the norm on the left-hand side corresponds to the trace on the right, which reduces it to a much easier question. For t positive but small there might be some messier behaviour. I don't know what the question means for $t < 0$. | |
| Feb 15, 2011 at 8:24 | comment | added | Pooja Singla | Thank you very much for your comment David. It is very useful for me. Would you suggest a reference for this result? I am also trying to understand the image of groups $1 + \pi^t \mathcal{O}_E$, $t \in \mathbb{Z}$ under the norm map in above situation. Do you have any comments for these groups? | |
| Feb 15, 2011 at 7:55 | vote | accept | Pooja Singla | ||
| Feb 14, 2011 at 15:49 | comment | added | David Loeffler | Just a remark: while (as the answers below show) it is not true that $[F^\times : N_{E/F} E^\times]$ has anything to do with ramification, it is true for any finite abelian extension of local fields that $[\mathcal{O}_F^\times : N_{E/F} \mathcal{O}_E^\times] = e_{E/F}$. | |
| Feb 14, 2011 at 13:37 | answer | added | David E Speyer | timeline score: 12 | |
| Feb 14, 2011 at 12:55 | comment | added | Pooja Singla | Thanks for you comment KConard. I have edited the question. | |
| Feb 14, 2011 at 12:54 | history | edited | Pooja Singla | CC BY-SA 2.5 |
added 13 characters in body
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| Feb 14, 2011 at 11:29 | answer | added | Chandan Singh Dalawat | timeline score: 7 | |
| Feb 14, 2011 at 11:09 | comment | added | KConrad | What does "some extension" mean? If you meant it to be a finite extension or an algebraic extension, please say so. Also, is the quadratic extension in the function field case supposed to be Galois or not? | |
| Feb 14, 2011 at 10:01 | history | edited | Andrey Rekalo | CC BY-SA 2.5 |
Typo in the title
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| Feb 14, 2011 at 8:54 | history | asked | Pooja Singla | CC BY-SA 2.5 |