Timeline for Class field towers
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 25, 2017 at 4:37 | comment | added | Ian Agol | BTW, it is not even known that there are infinitely many fields with class number 1. | |
| May 10, 2015 at 16:14 | comment | added | user6976 | @FranzLemmermeyer Your and David Loeffler's comments do help finding true motivation for the class tower problem. Thanks! | |
| May 9, 2015 at 15:04 | comment | added | Franz Lemmermeyer | I very much doubt that FLT would come into reach even if the class number of the Hilbert class field of cyclotomic fields always were 1: see mathoverflow.net/questions/13428/… | |
| Apr 23, 2015 at 15:05 | history | edited | user9072 |
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| Apr 23, 2015 at 14:50 | vote | accept | CommunityBot | moved from User.Id=6976 by developer User.Id=69903 | |
| Apr 23, 2015 at 14:03 | answer | added | Olivier | timeline score: 11 | |
| Apr 23, 2015 at 11:26 | comment | added | David Loeffler | If the class number of $\mathbf{Q}[\zeta]$ is coprime to $p$, where $\zeta$ is a primitive $p$-th root of unity and $p \ge 3$ is prime, then FLT for exponent $p$ follows. But this really needs control of the class group of the cyclotomic field itself; I can't see how knowing that $\mathbf{Q}[\zeta]$ embeds in some other larger field of class number 1 helps in any way. | |
| Apr 23, 2015 at 7:21 | comment | added | user6976 | @DavidLoeffler: If the class number of $\mathbb{Q}[\zeta]$ is 1, $\zeta^n=1$, then FLT for that $n$ follows, right? Isn't it enough to assume that $\zeta$ is inside a number field with class number 1? | |
| Apr 23, 2015 at 7:18 | comment | added | David Loeffler | Yes, because anything implies a true statement :-). Seriously, why should this question have any particular relation to FLT? | |
| Apr 23, 2015 at 7:15 | comment | added | user6976 | @DavidLoeffler: Would the case when $p$ is prime imply FLT? | |
| Apr 23, 2015 at 7:09 | comment | added | David Loeffler | The Golod--Shafarevich examples include cases where $K$ is imaginary quadratic, so $K$ is contained in $\mathbf{Q}(\zeta_n)$ for some suitable $n$; it follows that $\mathbf{Q}(\zeta_n)$ also has infinite class field tower. This doesn't deal with your more specific question about prime-order cyclotomic fields, though. | |
| Apr 23, 2015 at 6:28 | history | asked | user6976 | CC BY-SA 3.0 |