Timeline for Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2018 at 11:50 | comment | added | Watson | See also Kato, Saito, Number Theory 2 (2011), p. 167 (first paragraph). | |
| Jul 23, 2017 at 2:07 | history | edited | GH from MO |
edited tags
|
|
| Jan 15, 2010 at 2:44 | answer | added | KConrad | timeline score: 11 | |
| Jan 15, 2010 at 0:31 | comment | added | Emerton | @Ben: Since one doesn't a priori know that K/Q is abelian, there is no Artin map (a priori); one must argue with Frobenius elements and use Cebotarev density; see the edit to my answer. | |
| Jan 14, 2010 at 8:47 | vote | accept | CommunityBot | moved from User.Id=19475 by developer User.Id=69903 | |
| Jan 13, 2010 at 21:18 | comment | added | Kevin Buzzard | Here's a generalisation of this question. Say K is a Galois extension of Q, N is a positive integer, and a is an integer coprime to N. Say all primes congruent to a mod N split completely in K/Q. Is it true that all primes congruent to 1 mod N also split completely in K/Q? Probably a nifty application of Cebotarev will do it but I can't quite see it yet. | |
| Jan 13, 2010 at 21:06 | answer | added | Kevin Buzzard | timeline score: 13 | |
| Jan 13, 2010 at 21:01 | comment | added | Kevin Buzzard | Yes apologies. Doesn't a prime split completely in K iff it splits completely in the Galois closure? So I need all fields in my comment to be Galois, | |
| Jan 13, 2010 at 20:29 | answer | added | Emerton | timeline score: 10 | |
| Jan 13, 2010 at 20:19 | history | edited | Noah Snyder | CC BY-SA 2.5 |
Rewrote title to be a question. Added class-field-theory tag.
|
| Jan 13, 2010 at 20:12 | answer | added | user1073 | timeline score: 17 | |
| Jan 13, 2010 at 19:58 | comment | added | Dror Speiser | @buzzard: the "k contains a subfield..." only works when K is galois, and I am pretty sure the same holds for the first statement. | |
| Jan 13, 2010 at 19:48 | comment | added | Kevin Buzzard | A number field K is characterised up to isomorphism by the primes Spl(K) that split completely in it, if memory serves. Furthermore K contains a subfield isomorphic to L iff Spl(L) contains Spl(K) (up to a finite set of primes). So if all primes congruent to 1 mod N split completely in L, I think that's enough to prove that L is contained in Q(zeta_N). But that doesn't rule out a non-abelian extension of Q in which a prime splits iff it's, say, 3 mod 10. | |
| Jan 13, 2010 at 19:43 | history | asked | user19475 | CC BY-SA 2.5 |