| Title:
|
A determinant formula for the relative class number of an imaginary abelian number field (English) |
| Author:
|
Hirabayashi, Mikihito |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
22 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
133-140 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give a new formula for the relative class number of an imaginary abelian number field $K$ by means of determinant with elements being integers of a cyclotomic field generated by the values of an odd Dirichlet character associated to $K$. We prove it by a specialization of determinant formula of Hasse. (English) |
| Keyword:
|
imaginary abelian number field |
| Keyword:
|
relative class number |
| Keyword:
|
determinant |
| Keyword:
|
class number formula |
| MSC:
|
11R20 |
| MSC:
|
11R29 |
| idZBL:
|
Zbl 06410230 |
| idMR:
|
MR3303134 |
| . |
| Date available:
|
2015-01-27T09:37:01Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144126 |
| . |
| Reference:
|
[1] Girstmair, K.: The relative class numbers of imaginary cyclic fields of degrees 4, 6, 8 and 10.Math. Comp., 61, 1993, 881-887, Zbl 0787.11046, MR 1195428 |
| Reference:
|
[2] Hasse, H.: Über die Klassenzahl abelscher Zahlkörper.1952, Akademie-Verlag, Berlin, Reprinted with an introduction by J. Martinet, Springer Verlag, Berlin (1985). Zbl 0046.26003, MR 0842666 |
| Reference:
|
[3] Hirabayashi, M., Yoshino, K.: Remarks on unit indices of imaginary abelian number fields.Manuscripta math., 60, 1988, 423-436, Zbl 0654.12002, MR 0933473, 10.1007/BF01258662 |
| Reference:
|
[4] Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edition.1997, Springer Verlag, Berlin, MR 1421575 |
| Reference:
|
[5] Yamamura, K.: Bibliography on determinantal expressions of relative class numbers of imaginary abelian number fields.Number Theory. Dreaming in Dreams. Proceedings of the 5th China-Japan Seminar, 2010, 244-250, World Sci. Publ., Hackensack, Zbl 1202.11001, MR 2798466 |
| . |
