Timeline for Fermat over Number Fields
Current License: CC BY-SA 2.5
9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 30, 2010 at 21:07 | comment | added | Pete L. Clark | Let me say that it would make a nice, relatively short, project for someone (maybe a master's student?) to explicitly write out the connection between FLT(3) over quadratic fields and quadratic twists of a certain rational elliptic curve (with j = 0 ), including the connection between the class number condition and a Selmer group calculation. (To be sure, it seems likely that it has already been done, although I don't remember having seen it. But it would be worthwhile anyway, and would make for a very nice "here's why you should be interested in elliptic curves" talk.) | |
| Jan 30, 2010 at 21:02 | comment | added | Pete L. Clark | @Dror: To the best of my knowledge, no, it is not so easy. That was my point: the question whether F_3 has a nontrivial rational point over Q(\sqrt{m}) is equivalent to whether the quadratic twist of F_3 by m has positive rank. I believe that no explicit condition for the latter is known for any rational elliptic curve (including this one, which is simpler than usual in many respects because it has complex multiplication). | |
| Jan 30, 2010 at 20:40 | comment | added | Dror Speiser | Would a converse to the above be true as well? i.e. existence of a class of order 3 implies non trivial solution? If so, then this is quite more interesting than very difficult results in the theory of elliptic curves. I mean in this specific case, of course. | |
| Jan 30, 2010 at 20:27 | comment | added | Pete L. Clark | @DZ: You're absolutely right: I removed the offending sentence. By the way, if you had made this as a comment to my answer, I would have been notified of it automatically. | |
| Jan 30, 2010 at 19:51 | comment | added | Douglas Zare | Pete, despite your statement "So this is not so interesting," I maintain that for a fixed n, it is interesting to ask for which number fields there are solutions. | |
| Jan 30, 2010 at 9:13 | history | edited | Douglas Zare | CC BY-SA 2.5 |
Added reference.
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| Jan 30, 2010 at 8:33 | comment | added | Pete L. Clark | The Fermat cubic is an elliptic curve. Using known results in elliptic curve theory (specifically existence of quadratic twists with rank 0 and with rank 1, say), it follows easily that there will be infinitely many quadratic fields over which the Fermat cubic has only the trivial solutions and infinitely many more over which it has nontrivial solutions. | |
| Jan 30, 2010 at 7:41 | history | answered | Douglas Zare | CC BY-SA 2.5 |