Timeline for A special integral polynomial
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 13, 2010 at 21:45 | comment | added | Pete L. Clark | By the way, since this is an application of the sort of weak approximation + Krasner argument that I have used in my own work and now introduced in my course on local fields, having looked at Keith's paper it's natural to try to make a similar argument work to get the alternating groups A_n as Galois groups over any global field. [Yes, I know this is due to Hilbert.] But it's not immediately clear to me how to do it -- can anyone help? | |
| Mar 13, 2010 at 21:23 | comment | added | Kevin Buzzard | :-/ Now you mention the result I remember using it about 10 years ago to check that the char poly of T_2 on S_k(SL_2(Z)) had Galois group S_n for all k<=2048. Daft story connected with this: after I checked this I emailed William Stein telling him what I had done, and the next day he emailed me back saying he'd just done k=2050 so now he held the record :-) | |
| Mar 13, 2010 at 20:22 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 339 characters in body
|
| Mar 13, 2010 at 20:04 | comment | added | Pete L. Clark | @Kevin: Thanks for the comment. I completed the argument along the lines you suggested. | |
| Mar 13, 2010 at 19:56 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 348 characters in body
|
| Mar 13, 2010 at 18:33 | comment | added | Kevin Buzzard | @Pete: I'm sure this works but I don't quite understand the argument yet. Your strategy shows that I can find f with no real roots and such that for some finite set of primes p in S, f mod p factors in a given way. The upshot is that you can decree the cycle type of Frob_p for p in a finite set. But you can't control which roots are in which cycle, can you? So aren't you left with the following issue: you have to prove that if G is a transitive subgroup of S_{2n} containing an element of each cycle type, then G=S_{2n}. No doubt this is standard but don't you need it to complete the argument? | |
| Mar 13, 2010 at 18:04 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 39 characters in body; added 7 characters in body
|
| Mar 13, 2010 at 17:57 | history | answered | Pete L. Clark | CC BY-SA 2.5 |