Timeline for Expressing a number field as a composite of extensions ramified at one place
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 13, 2010 at 23:05 | vote | accept | Kevin Ventullo | ||
| Nov 13, 2010 at 23:04 | comment | added | Kevin Ventullo | @Tim: Nice answer (both of them)! That second one is quite slick. | |
| Nov 13, 2010 at 22:48 | comment | added | Tim Dokchitser | You're right of course, I did not put this correctly! Also, as my brother has pointed out, all this representation theory nonsense can be avoided: the group $G=G_1*...*G_n$ acts on three points (the embeddings of $K$ into $C$), and its very difficult to have pairwise commuting groups to act on 3 points. As one of them, corresponding to $p=3$, is the whole of $S_3$, all the others must act trivially, so $K$ cannot be ramified at any other prime. | |
| Nov 13, 2010 at 21:26 | comment | added | BCnrd | Dear Tim: Very nice. A tiny correction is that pairwise triviality of intersections does not quite suffice to justify the required linear disjointness in general (but we have more ramification information, and so have disjointness of each $L_i$ against the compositum of the others, say also proceeding by induction on the number of $L_i$ or whatever; of course, you had all of this in mind so as to not really need to worry about it). For example, in a biquadratic field the 3 quadratic subfields have pairwise trivial intersections but the triple is not a linearly disjoint collection. | |
| Nov 13, 2010 at 19:40 | history | answered | Tim Dokchitser | CC BY-SA 2.5 |