Abstract
Let K = F(T) be the rational function field over a finite field of q elements. For any polynomial f(T) ∈ F [T] with positive degree, denote by Λ f the torsion points of the Carlitz module for the polynomial ring F[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(Λ P ) of degree k over F(T), where P ∈ F[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.
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This work was supported by NSFC (No. 11301071), CPSF (No. 2013M531244) and NSFJ (No. 1202101c).
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Zhao, Z., Wu, X. On the subfields of cyclotomic function fields. Czech Math J 63, 799–803 (2013). https://doi.org/10.1007/s10587-013-0053-x
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DOI: https://doi.org/10.1007/s10587-013-0053-x