On the subfields of cyclotomic function fields

Zhao, Zhengjun, and Wu, Xia. "On the subfields of cyclotomic function fields." Czechoslovak Mathematical Journal 63.3 (2013): 799-803. <http://eudml.org/doc/260665>.

@article{Zhao2013,
abstract = {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.},
author = {Zhao, Zhengjun, Wu, Xia},
journal = {Czechoslovak Mathematical Journal},
keywords = {cyclotomic function fields; $L$-function; class number formula; cyclotomic function fields; -function; class number formula},
language = {eng},
number = {3},
pages = {799-803},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the subfields of cyclotomic function fields},
url = {http://eudml.org/doc/260665},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Zhao, Zhengjun
AU - Wu, Xia
TI - On the subfields of cyclotomic function fields
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 799
EP - 803
AB - 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.
LA - eng
KW - cyclotomic function fields; $L$-function; class number formula; cyclotomic function fields; -function; class number formula
UR - http://eudml.org/doc/260665
ER -