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Class number formulae in the form of a product of determinants in function fields

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  09 April 2009

Jaehyun Ahn ,
Soyoung Choi  and
Hwanyup Jung
Jaehyun Ahn
Affiliation:
Department of MathematicsKAIST Daejon 305-701Korea e-mail: jaehyun@math.kaist.ac.kryoung@math.kaist.ac.kr
Soyoung Choi
Affiliation:
Department of MathematicsKAIST Daejon 305-701Korea e-mail: jaehyun@math.kaist.ac.kryoung@math.kaist.ac.kr
Hwanyup Jung
Affiliation:
Department of Mathematics EducationChungbuk National UniversityCheongju Chungbuk 361-763Korea e-mail: hyjung@chungbuk.ac.kr
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Abstract

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In this paper, we generalize the Kučera's group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an intermediate field in the tower of cyclotomic function fields with prime power conductor.

Keywords

cyclotomic function fieldsclass number

MSC classification

Secondary: 11R60: Cyclotomic function fields (class groups, Bernoulli objects, etc.) 11R58: Arithmetic theory of algebraic function fields

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 2 , April 2005 , pp. 227 - 238
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bae, S., Jung, H. and Ahn, J., ‘Determinant formulas for class numbers in function fields’, Math. Comp. 74 (2005), 935–965.Google Scholar
[2]Bae, S. and Kang, P.-L., ‘Class numbers of cyclotomic function fields’, Acta Arith. 102 (2002), 251–259.CrossRefGoogle Scholar
[3]Hayes, D. R., ‘Explicit class field theory for rational function fields’, Trans. Amer. Math. Soc. 189 (1974), 77–91.CrossRefGoogle Scholar
[4]Hirabayashi, M., ‘A generalization of Maillet and Demyanenko determinants for the cyclotomic Zp-extension’, Abh. Math. Sem. Univ. Hamburg 71 (2001), 15–27.CrossRefGoogle Scholar
[5]Jung, H. and Ahn, J., ‘Demjanenko matrix and recursion formula for relative class number over function fields’, J. Number Theory 98 (2003), 55–66.CrossRefGoogle Scholar
[6]Kučera, R., ‘Formulae for the relative class number of an imaginary abelian field in the form of a product of determinants’, Acta Math. Inform. Univ. Ostraviensis 10 (2002), 79–83.Google Scholar
[7]Nathanson, M. B., Elementary methods in number theory, Graduate Texts in Mathematics 195 (Springer, New York, 2000).Google Scholar
[8]Rosen, M., ‘A note on the relative class number in function fields’, Proc. Amer. Math. Soc. 125 (1997), 1299–1303.CrossRefGoogle Scholar
[9]Rosen, M., Number theory in function fields, Graduate Texts in Mathematics 210 (Springer, New York, 2002).Google Scholar
[10]Tsumura, H., ‘A note on the Demjanenko matrices related to the cyclotomic Zp–extension’, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 99–103.CrossRefGoogle Scholar