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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on the relative class number
in function fields

Author: Michael Rosen
Journal: Proc. Amer. Math. Soc. 125 (1997), 1299-1303
MSC (1991): Primary 11R29; Secondary 11R58, 14H05
MathSciNet review: 1371139
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Abstract: Let $F$ be a finite field, $A=F[T]$, and $k=F(T)$. Let $K_{m}=k(\Lambda _{m})$ be the field extension of $k$ obtained by adjoining the $m$-torsion on the Carlitz module. The class number $h_{m}$ of $K_{m}$ can be written as a product $h_{m}=h_{m}^{+}h_{m}^{-}$. The number $h_{m}^{-}$ is called the relative class number. In this paper a formula for $h_{m}^{-}$ is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of $p$-th roots of unity. Some consequences of this formula are also derived.

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Additional Information

Michael Rosen
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912-0001

Received by editor(s): July 2, 1995
Received by editor(s) in revised form: November 15, 1995
Additional Notes: This work was partially supported with a grant from the National Science Foundation.
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society