Timeline for Can algebraic number fields be generalized in a similar way to function fields in 1 variable over a finite field?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 28, 2010 at 14:09 | vote | accept | teil | ||
| Mar 22, 2010 at 1:55 | comment | added | Joel Dodge | A little comment on your question: Studying varieties of dimension greater than 1 is not the same thing as studying their function fields. | |
| Mar 21, 2010 at 14:36 | comment | added | dke | In fact, this question put me in mind of Kato's survey "Generalized class field theory" from the 1990 ICM, since in the introduction, Kato in his own inimitable way explains how human beings need to start doing arithmetic in fields which are finitely generated over their prime subfields rather than merely in the one-dimensional classical global fields. | |
| Mar 21, 2010 at 14:33 | comment | added | dke | Whilst the standard global fields are special in many ways, there is actually a well-developed class field theory for these higher-dimensional fields, with both idelic and ideal-theoretic descriptions due to Parshin, Kato, Saito in the 80s and many others since. | |
| Mar 21, 2010 at 12:58 | comment | added | Regenbogen | There is an axiomatic characterization of global fields by Artin and Whaples. Classfield theory as done by Chevalley can be carried out provided these axioms are true. Therefore number fields and function fields in one variable are special. I do not think classfield theory can be carried out for function fields in more than one variable, for instance. | |
| Mar 21, 2010 at 9:30 | answer | added | Pete L. Clark | timeline score: 5 | |
| Mar 21, 2010 at 8:54 | history | asked | teil | CC BY-SA 2.5 |