Timeline for Interpolation of families of local fields
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 24, 2018 at 6:27 | comment | added | Aurel | @JulianRosen Good point! | |
| Sep 23, 2018 at 12:56 | answer | added | François Brunault | timeline score: 3 | |
| Sep 23, 2018 at 6:12 | history | undeleted | user129156 | ||
| Sep 23, 2018 at 2:49 | history | deleted | user129156 | via Vote | |
| Sep 22, 2018 at 23:53 | history | edited | user129156 | CC BY-SA 4.0 |
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| Sep 22, 2018 at 22:51 | comment | added | Julian Rosen | @Aurel If $K$ is not Galois over $\mathbb{Q}$, then different primes of $K$ over the same prime of $\mathbb{Q}$ can have non-isomorphic completions, so $K$ can correspond to uncountably many sequences $(K_v)$. | |
| Sep 22, 2018 at 21:59 | comment | added | KConrad | While there are some similarities with the author of the handout, that's not me. | |
| Sep 22, 2018 at 21:07 | history | edited | user129156 | CC BY-SA 4.0 |
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| Sep 22, 2018 at 21:04 | comment | added | user129156 | @KConrad Thanks. Your handout answers my question if $\Sigma$ is replaced by a finite subset | |
| Sep 22, 2018 at 21:01 | history | edited | user129156 | CC BY-SA 4.0 |
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| Sep 22, 2018 at 21:00 | comment | added | user129156 | @KConrad I’m studying Artin and Tate’s book on class field theory. I’m trying myself on this kind of questions | |
| Sep 22, 2018 at 20:57 | comment | added | David Lampert | And having allowable densities of splitting types isn't sufficient e.g. if half $K_\nu$ have degree=1 and half have degree=2 then the specific halves are governed by class field theory, and it seems very hard to give other restrictions for higher degrees. | |
| Sep 22, 2018 at 20:52 | comment | added | KConrad | Are you asking these questions on MO recently for a specific reason, or out of idle curiosity? For a description of some known results about realizing a finite set of local fields as completions of a single number field, take a look at math.stanford.edu/~conrad/248APage/handouts/localglobal.pdf. | |
| Sep 22, 2018 at 20:48 | comment | added | KConrad | No. Suppose $\Omega$ is all but one nonarchimedean place of $\mathbf Q$. If there were such a $K$ then all but one prime splits completely in $K$, so the density you ask about is 1 and that implies $K = \mathbf Q$: the density of primes that split completely in $K$ is $1/[\widetilde{K}:\mathbf Q]$ where $\widetilde{K}$ is the Galois closure of $K$ over $\mathbf Q$. | |
| Sep 22, 2018 at 20:46 | comment | added | Aurel | There are uncountably many such collections, but there are only countably many number fields. | |
| Sep 22, 2018 at 20:23 | history | asked | user129156 | CC BY-SA 4.0 |