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On real quadratic function fields
of Chowla type with ideal class number one


Authors: Keqin Feng and Weiqun Hu
Journal: Proc. Amer. Math. Soc. 127 (1999), 1301-1307
MSC (1991): Primary 11R11, 11R29
DOI: https://doi.org/10.1090/S0002-9939-99-05004-2
Published electronically: January 27, 1999
MathSciNet review: 1622805
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathbb{F}_q$ be the finite field with $q$ elements, (2${\not|}q$), $k= \mathbb{F}_q(x)$, $K=k(\sqrt{D})$ where $D=D(x) =A(x)^2+a$ is a square-free polynomial in $\mathbb{F}_q[x]$ with $\deg A(x)\geq 1$ and $a\in \mathbb{ F}_q^*$. In this paper several equivalent conditions for the ideal class number $h(O_K)$ to be one are presented and all such quadratic function fields with $h(O_K)=1$ are determined.


References [Enhancements On Off] (What's this?)

  • 1. E.Artin, Quadratische Körper in Gebiet der höhren Kongruenzen I,II, Math. Zeit. 19(1924),154-246.
  • 2. M.Deuring, Lectures on the theory of algebraic functions of one variable, LN in Math,NO.314,Springer-Verlag,1973. MR 49:8970
  • 3. Keqin Feng and Shuling Sun, On class number of quadratic fields, Proceeding of First International Symposium on Algebraic Structures and Number Theory (1988,Hong Kong),Edited by S.P.Lam and K.P.Shum, World Scientific, 1990,pp. 88-133. MR 91m:11098
  • 4. Weiqun Hu, On the Minkowski constant for function fields, preprint, 1997.
  • 5. J.R.C.Leitzel,M.L.Madan and C.S.Queen, Algebraic function fields with small class number,Jour. of Number Theory, 7(1975),11-27. MR 51:5561
  • 6. R.E.MacRae, On unique factorization in certain rings of algebraic functions, Jour. of Algebra, 17(1971),243-261. MR 42:7643
  • 7. M.L.Madan and C.S.Queen, Algebraic function fields of class number one, Acta Arith. 20(1972),423-432. MR 46:5287
  • 8. R.A. Mollin, Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S.Chowla,Proc. Amer. Math. Soc. 102(1988),17-21. MR 89d:11098
  • 9. R.A. Mollin, and H.C.Williams, A conjecture of S.Chowla via the generalized Riemann Hypothesis, Ibid,102(1988),794-796. MR 89d:11090
  • 10. Xianke Zhang, Ambiguous classes and 2-rank of class group of quadratic function field,Jour. of China Univ. of Sci. & Tech.,19(1987), 425-431. MR 89j:11115

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Additional Information

Keqin Feng
Affiliation: Graduate School at Beijing, University of Science and Technology of China, P. O. Box 3908, Beijing 100039, People’s Republic of China

Weiqun Hu
Affiliation: The Fundamental Science Department, Nanjing Agriculture College, Nanjing 210038, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-99-05004-2
Keywords: Quadradic field, function field, class number
Received by editor(s): August 20, 1997
Published electronically: January 27, 1999
Additional Notes: Research supported by the Natural Science Foundation and the National Educational Committee of China.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society

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