Geodesic
On the subfields of cyclotomic function fields
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 799-803
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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.
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.
DOI : 10.1007/s10587-013-0053-x
Classification : 11R18, 11R58, 11R60
Keywords: cyclotomic function fields; $L$-function; class number formula 
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Zhao, Zhengjun; Wu, Xia. On the subfields of cyclotomic function fields. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 799-803. doi: 10.1007/s10587-013-0053-x

[1] Bae, S., Lyun, P.-L.: Class numbers of cyclotomic function fields. Acta. Arith. 102 (2002), 251-259. | DOI | MR | Zbl

[2] Galovich, S., Rosen, M.: Units and class groups in cyclotomic function fields. J. Number Theory 14 (1982), 156-184. | DOI | MR | Zbl

[3] Guo, L., Linghsuen, S.: Class numbers of cyclotomic function fields. Trans. Am. Math. Soc. 351 (1999), 4445-4467. | DOI | MR

[4] Hayes, D. R.: Analytic class number formulas in global function fields. Invent. Math. 65 (1981), 49-69. | DOI | MR

[5] Rosen, M.: Number Theory in Function Fields. Graduate Texts in Mathematics 210. Springer, New York (2002). | MR | Zbl

[6] Rosen, M.: The Hilbert class field in function fields. Expo. Math. 5 (1987), 365-378. | MR | Zbl

[7] Zhao, Z. Z.: The Arithmetic Problems of Some Special Algebraic Function Fields. Ph.D. Thesis, NJU (2012), Chinese.

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