Timeline for An alternative description of K^*/Nm(L^*)
Current License: CC BY-SA 2.5
9 events
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| Sep 28, 2010 at 18:52 | comment | added | Keenan Kidwell | @Evgeny Regarding your edit, yes, when $L/K$ is a finite cyclic extension with group $G$, $H^0_T(G,L^\times)$ ($0$-dimensional Tate cohomology), your norm residue group, is isomorphic to $H^2(G,L^\times)$ via cup product with a generator of $H^0(G,\mathbb{Z})$, which is cyclic of order $\vert G\vert$ (this isomorphism depends on the choice of generator). | |
| Sep 28, 2010 at 15:11 | history | edited | Evgeny Shinder | CC BY-SA 2.5 |
added 181 characters in body; added 8 characters in body
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| Sep 28, 2010 at 4:20 | answer | added | Franz Lemmermeyer | timeline score: 4 | |
| Sep 28, 2010 at 1:45 | answer | added | Alex B. | timeline score: 7 | |
| Sep 27, 2010 at 23:10 | comment | added | Dror Speiser | There are very simple cases where "explicit" is really explicit. The simplest being the trivial case of a quadratic extension of the rationals: the group is isomorphic to the product (strict class group)x(infinitely many copies of $C_2$), a copy for each inert prime. A generalization works for Kummer extensions as well. So not all is lost. | |
| Sep 27, 2010 at 22:44 | comment | added | BCnrd | The meaning of "nice explicit" is unclear; hard to know what you have in mind to do with an answer. In the cyclic Galois case for number fields one can give a pseudo-answer using class field theory and general considerations with cohomology of tori (see case $r=3$ on p. 199 of Cassels-Frohlich, and use double-periodicity of cyclic cohomology to shift to $r=1$), but not really "explicit", is it? With general fields it seems doubtful that anything useful can be said. | |
| Sep 27, 2010 at 21:55 | history | edited | Evgeny Shinder | CC BY-SA 2.5 |
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| Sep 27, 2010 at 21:52 | comment | added | Martin Brandenburg | If $L/K$ is cyclic with generator $\sigma$, then Hilbert 90 tells us that $a \mapsto a \sigma(a)^{-1}$ yields an isomorphism $L^* / K^* \cong ker(N)$. But I don't see any connection with $coker(N)$. | |
| Sep 27, 2010 at 21:19 | history | asked | Evgeny Shinder | CC BY-SA 2.5 |