충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제23권4호
  • /
  • Pages.699-704
  • /
  • 2010
  • /
  • 1226-3524(pISSN)
  • /
  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

HILBERT 2-CLASS FIELD TOWERS OF IMAGINARY QUADRATIC FUNCTION FIELDS

  • Ahn, Jaehyun (Department of Mathematics Chungnam National University) ;
  • Jung, Hwanyup (Department of Mathematics Education Chungbuk National University)
  • 투고 : 2010.07.16
  • 심사 : 2010.11.09
  • 발행 : 2010.12.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we prove that the Hilbert 2-class field tower of an imaginary quadratic function field $F=k({\sqrt{D})$ is infinite if $r_2({\mathcal{C}}(F))=4$ and exactly one monic irreducible divisor of D is of odd degree, except for one type of $R{\acute{e}}dei$ matrix of F. We also compute the density of such imaginary quadratic function fields F.

키워드

참고문헌

  1. S. Bae and H. Jung, On Hilbert 2-class field towers of quadratic function fields, preprint.
  2. E. Benjamin, On imaginary quadratic number fields with 2-class group of rank 4 and infinite 2-class field tower, Pacific J. Math. 201 (2001), no. 2, 257-266. https://doi.org/10.2140/pjm.2001.201.257
  3. E. Benjamin,On a question of Martinet concerning the 2-class field tower of imaginary quadratic number field, Ann. Sci. Math. Quebec 26 (2002), no. 1, 1-13.
  4. H. Jung, Density of class groups of imaginary $\ell$-cyclic function fields, submitted for publication.
  5. J. Martinet, Tours de corps de classes et estimations de discriminants, Invent. Math. 44 (1978), no. 1, 65-73. https://doi.org/10.1007/BF01389902
  6. M. Rosen, The Hilbert class field in function fields, Exposition. Math. 5 (1987), no. 4, 365-378.
  7. R. Schoof, Algebraic curves over $F_{2}$ with many rational points, J. Number Theory 41 (1992), no. 1, 6-14. https://doi.org/10.1016/0022-314X(92)90079-5
  8. Y. Sueyoshi, Infinite 2-class field towers of some imaginary quadratic number fields, Acta. Arith. 113 (2004), 251-257. https://doi.org/10.4064/aa113-3-3
  9. C. Wittmann, $\ell$-Class groups of cyclic function fields of degree $\ell$, Finite fields and their applications 13 (2007), 327-347. https://doi.org/10.1016/j.ffa.2005.09.001
  10. C. Wittmann, Densities for 4-ranks of quadratic function fields, J. Number Theory 129 (2009), 2635-2645. https://doi.org/10.1016/j.jnt.2009.03.004