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A note on the relative class number in function fields

Authors:	Michael Rosen

Institution:	Department of Mathematics, Brown University, Providence, Rhode Island 02912-0001

Abstract:	Let be a finite field, , and . Let $K_{m}=k(\Lambda _{m})$ be the field extension of obtained by adjoining the -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 -th roots of unity. Some consequences of this formula are also derived.

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