Timeline for Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| May 25, 2010 at 11:04 | comment | added | David E Speyer | If your students already believe in unique factorization into ideals, and that $\mathbb{Z}[zeta_n]$ is integrally closed, it is easy enough to show that $Z(s)$ is the zeta function of the cyclotomic field. Just analyze $\mathbb{Z}[zeta_n]/p \cong \mathbb{F}_p[t]/t^n-1$. (This is a bit easier for $p$ not dividing $n$.) But, certainly, this will be very difficult in an analytically focused class. For that, I would go to Serre's proof. | |
| May 25, 2010 at 11:00 | comment | added | David E Speyer | Of course, it is true that the quadratic case is harder than the others. But I think that, pedagogically and aesthetically, it is better to give a proof which handles the hardest case in such a way that the other cases are covered at the same time. When I am trying to learn a proof, I find that every case is another burden on my understanding. | |
| May 25, 2010 at 10:56 | comment | added | David E Speyer | I'm not sure why the class number formula for arbitrary number fields is harder than the quadratic case. The presence of an infinite unit group makes the proof harder, but you already have to deal with real quadratic fields. The differences between a rank one unit group and a general one are only notational. (Although, depending on your audience, notation can be a big savings.) | |
| May 25, 2010 at 10:53 | comment | added | David E Speyer | I'm kind of surprised that this was the highest rated answer, since it is more of an answer to the most elegant proof than the shortest proof. But I will reply to some of your comments: | |
| May 25, 2010 at 5:24 | comment | added | Junkie | I guess I can re-phrase my comment as: if you are going to use the class number formula in the end, it seems an excess to do so with cyclotomic fields. As Robin Chapman noted, you can just pair off complex conjugate characters quite easily, and then are left with just quadratics for the class number formula (which are easier from a ground-up viewpoint, though for someone facile in number fields, it matters not I guess). Of course, this is notably opposite to the what the OP wanted. | |
| May 25, 2010 at 5:10 | comment | added | Junkie | First a quibble: is it common to take the $\zeta$-function of a cyclotomic ring, rather than a cyclotomic field? And #1 should say "ideals of Z[zeta_n]"? Is it that easy to declare (to a student) that $Z(s)$ is indeed the $\zeta$-function of the cyclotomic field -- coming from an analytic number theory background (Ayoub's book from the 60s for me actually), Kronecker-Weber, or whatever Lang would have in his book, might end up being a side-trip if this is not obvious. For that matter, discreteness of the unit group (as compared to class no formula for quadratic chars) is also overhead. | |
| May 24, 2010 at 20:49 | history | answered | David E Speyer | CC BY-SA 2.5 |