| Title:
|
On the subfields of cyclotomic function fields (English) |
| Author:
|
Zhao, Zhengjun |
| Author:
|
Wu, Xia |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
799-803 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $K = \mathbb {F}_q(T)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f(T)\in \mathbb {F}_q[T]$ with positive degree, denote by $\Lambda _f$ the torsion points of the Carlitz module for the polynomial ring $\mathbb {F}_q[T]$. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K(\Lambda _P)$ of degree $k$ over $\mathbb {F}_q(T)$, where $P\in \mathbb {F}_q[T]$ is an irreducible polynomial of positive degree and $k>1$ is a positive divisor of $q-1$. A formula for the analytic class number for the maximal real subfield $M^+$ of $M$ is also presented. Futhermore, a relative class number formula for ideal class group of $M$ will be given in terms of Artin $L$-function in this paper. (English) |
| Keyword:
|
cyclotomic function fields |
| Keyword:
|
$L$-function |
| Keyword:
|
class number formula |
| MSC:
|
11R18 |
| MSC:
|
11R58 |
| MSC:
|
11R60 |
| idZBL:
|
Zbl 06282111 |
| idMR:
|
MR3125655 |
| DOI:
|
10.1007/s10587-013-0053-x |
| . |
| Date available:
|
2013-10-07T12:09:20Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143490 |
| . |
| Reference:
|
[1] Bae, S., Lyun, P.-L.: Class numbers of cyclotomic function fields.Acta. Arith. 102 (2002), 251-259. Zbl 0989.11064, MR 1884718, 10.4064/aa102-3-4 |
| Reference:
|
[2] Galovich, S., Rosen, M.: Units and class groups in cyclotomic function fields.J. Number Theory 14 (1982), 156-184. Zbl 0483.12003, MR 0655724, 10.1016/0022-314X(82)90045-2 |
| Reference:
|
[3] Guo, L., Linghsuen, S.: Class numbers of cyclotomic function fields.Trans. Am. Math. Soc. 351 (1999), 4445-4467. MR 1608317, 10.1090/S0002-9947-99-02325-9 |
| Reference:
|
[4] Hayes, D. R.: Analytic class number formulas in global function fields.Invent. Math. 65 (1981), 49-69. MR 0636879, 10.1007/BF01389294 |
| Reference:
|
[5] Rosen, M.: Number Theory in Function Fields.Graduate Texts in Mathematics 210. Springer, New York (2002). Zbl 1043.11079, MR 1876657 |
| Reference:
|
[6] Rosen, M.: The Hilbert class field in function fields.Expo. Math. 5 (1987), 365-378. Zbl 0632.12017, MR 0917350 |
| Reference:
|
[7] Zhao, Z. Z.: The Arithmetic Problems of Some Special Algebraic Function Fields.Ph.D. Thesis, NJU (2012), Chinese. |
| . |
