Timeline for Avoiding Minkowski's theorem in algebraic number theory.
Current License: CC BY-SA 2.5
8 events
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| Nov 2, 2012 at 0:43 | comment | added | Bruno Stonek | Dear KConrad, please see this related question in math.SE math.stackexchange.com/questions/227201/… . Thanks. | |
| Jun 17, 2010 at 1:51 | comment | added | KConrad | That nobody read Kronecker's thesis isn't a complete explanation for why he doesn't get credit on this bound related to the class group. Kronecker reproduces the argument in his long paper on arithmetic in polynomial rings (Crelle 92 1882, see pp. 64--65) and points out there that the basic idea was already in his thesis. Nobody at the time understood this paper very well, so one should also say Kronecker doesn't get the credit because his 1882 paper was not widely read either. (In Dedekind's XI-th supplement, 4th ed., Sect. 181, Kronecker's argument is used without attribution.) | |
| Mar 23, 2010 at 15:52 | comment | added | KConrad | I found the thesis (Crelle vol. 93 pp. 1--52, or visible at the link gdz.sub.uni-goettingen.de/dms/load/img/…) and on page 15 the "Hurwitz" argument jumps out from the Latin. It looks like here Kronecker is working in a subfield (with degree $\lambda$) of $\mathbf Q(\zeta_p)$ where $p$ is prime, rather than in a general cyclotomic field. | |
| Mar 23, 2010 at 6:12 | comment | added | Franz Lemmermeyer | Probably because hardly anyone ever read Kronecker's thesis. It is in Latin, and his main result is "Dirichlet's unit theorem" for cyclotomic fields. | |
| Mar 22, 2010 at 19:50 | comment | added | KConrad | Aha! Saying this proof of class number finiteness goes back to Hurwitz is repeated in another book also. If it is originally due to Kronecker, do you know why it is attributed to Hurwitz? | |
| Mar 22, 2010 at 19:27 | comment | added | Franz Lemmermeyer | The proof in Ireland-Rosen is essentially due to Kronecker (his thesis), and predates even the introduction of ideal numbers. Kronecker gave his proof in the case of cyclotomic fields; the proof goes through in general, however, once you know what an integral basis is. | |
| Mar 22, 2010 at 18:31 | history | edited | KConrad | CC BY-SA 2.5 |
added 28 characters in body; added 373 characters in body
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| Mar 22, 2010 at 17:09 | history | answered | KConrad | CC BY-SA 2.5 |