대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제27권3호
  • /
  • Pages.455-464
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  • 2012
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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ON THE RELATIVE ZETA FUNCTION IN FUNCTION FIELDS

  • 투고 : 2011.02.02
  • 발행 : 2012.07.31
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초록

In the previous paper [8], the author gave a determinant formula of relative zeta function for cyclotomic function fields. Our purpose of this paper is to extend this result for more general function fields. As an application of our formula, we will give determinant formulas of class numbers for constant field extensions.

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참고문헌

  1. J. Ahn, S. Choi, and H. Jung, Class number formulae in the form of a product of determinants in function fields, J. Aust. Math. Soc. 78 (2005), no. 2, 227-238. https://doi.org/10.1017/S1446788700008053
  2. L. Carlitz and F. R. Olson, Maillet's determinant, Proc. Amer. Math. Soc. 6 (1955), 265-269.
  3. K. Girstmair, The relative class numbers of imaginary cyclic fields of degrees 4, 6, 8, and 10, Math. Comp. 61 (1993), no. 204, 881-887.
  4. D. R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77-91. https://doi.org/10.1090/S0002-9947-1974-0330106-6
  5. T. Metsankyla, Bemerkungen uber den ersten Faktor der Klassenzahl des Kreiskorpers, Ann. Univ. Turku. Ser. A I No. 105 (1967), 15 pp.
  6. M. Rosen, A note on the relative class number in function fields, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1299-1303. https://doi.org/10.1090/S0002-9939-97-03748-9
  7. M. Rosen, Number Theory in Function Fields, Springer-Verlag, Berlin, 2002.
  8. D. Shiomi, A determinant formula for relative congruence zeta functions for cyclotomic function fields, J. Aust. Math. Soc. 89 (2010), no. 1, 133-144. https://doi.org/10.1017/S1446788710000261
  9. K. Tateyama, Maillet's determinant, Sci. Papers College Gen. Ed. Univ. Tokyo 32 (1982), no. 2, 97-100.
  10. L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982.