Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Class groups of global function fields with certain splitting behaviors of the infinite prime


Author: Yoonjin Lee
Journal: Proc. Amer. Math. Soc. 137 (2009), 415-424
MSC (2000): Primary 11R29; Secondary 11R58
DOI: https://doi.org/10.1090/S0002-9939-08-09581-6
Published electronically: October 6, 2008
MathSciNet review: 2448559
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For certain two cases of splitting behaviors of the prime at infinity with unit rank $ r$, given positive integers $ m, n$, we construct infinitely many global function fields $ K$ such that the ideal class group of $ K$ of degree $ m$ over $ \mathbb{F}(T)$ has $ n$-rank at least $ m-r-1$ and the prime at infinity splits in $ K$ as given, where $ \mathbb{F}$ denotes a finite field and $ T$ a transcendental element over $ \mathbb{F}$. In detail, for positive integers $ m$, $ n$ and $ r$ with $ 0 \le r \le m-1$ and a given signature $ (e_i, \mathfrak{f}_i)$, $ 1 \le i \le r+1$, such that $ \sum_{i=1}^{r+1}{e_i\mathfrak{f}_i} =m$, in the following two cases where $ e_i$ is arbitrary and $ \mathfrak{f}_i =1$ for each $ i$, or $ e_i =1$ and $ \mathfrak{f}_i$'s are the same for each $ i$, we construct infinitely many global function fields $ K$ of degree $ m$ over $ \mathbb{F}(T)$ such that the ideal class group of $ K$ contains a subgroup isomorphic to $ (\mathbb{Z}/n\mathbb{Z})^{m-r-1}$ and the prime at infinity $ {\wp_\infty}$ splits into $ r+1$ primes $ \mathfrak{P}_1, \mathfrak{P}_2, \cdots, \mathfrak{P}_{r+1}$ in $ K$ with $ e(\mathfrak{P}_i/{\wp_\infty}) = e_i$ and $ \mathfrak{f}(\mathfrak{P}_i/{\wp_\infty}) = \mathfrak{f}_i$ for $ 1 \le i \le r+1$ (so, $ K$ is of unit rank $ r$).


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

  • 1. T. Azuhata and H. Ichimura, On the divisibility problem of the class numbers of algebraic number fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), 579-585. MR 731519 (85a:11021)
  • 2. C. Friesen, Class number divisibility in real quadratic function fields, Canad. Math. Bull. 35 (1992), 361-370. MR 1184013 (93h:11130)
  • 3. A. Frohlich and M.J. Taylor, Algebraic Number Theory, Cambridge University Press, 1993.
  • 4. S. Lang, Algebra, 2nd edition, Addison-Wesley, Reading, MA, 1984. MR 0197234 (33:5416)
  • 5. Y. Lee, The structure of the class groups of global function fields with any unit rank, J. Ramanujan Math. Soc. 20, No. 2 (2005), 125-145. MR 2169092 (2007m:11154)
  • 6. Y. Lee and A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005), 2883-2889. MR 2159765 (2006e:11177)
  • 7. Y. Lee and A. Pacelli, Higher rank subgroups in the class groups of imaginary function fields, J. Pure Appl. Algebra 207 (2006), 51-62. MR 2244260 (2007d:11124)
  • 8. T. Nagell, Uber die Klassenzahl imaginar quadratischer Zahlkorper, Abh. Math. Sem. Univ. Hamburg 1 (1922), 140-150.
  • 9. S. Nakano, On ideal class groups of algebraic number fields, J. Reine Angew. Math. 358 (1985), 61-75. MR 797674 (86k:11063)
  • 10. J. Neukirch, Algebraic Number Theory, Springer-Verlag, Berlin, 1999. MR 1697859 (2000m:11104)
  • 11. A. Pacelli, Abelian subgroups of any order in class groups of global function fields, J. Number Theory 106 (2004), 26-49. MR 2029780 (2004m:11193)
  • 12. A. Pacelli, The prime at infinity and the rank of the class group in global function fields, J. Number Theory 116 (2006), 311-323. MR 2195928 (2006k:11228)
  • 13. M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York, 2002. MR 1876657 (2003d:11171)
  • 14. Y. Yamamoto, On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), 57-76. MR 0266898 (42:1800)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11R29, 11R58

Retrieve articles in all journals with MSC (2000): 11R29, 11R58


Additional Information

Yoonjin Lee
Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 120-750, Republic of Korea
Email: yoonjinl@ewha.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-08-09581-6
Keywords: Class group, class number, rank of class group, imaginary function field
Received by editor(s): April 26, 2007
Published electronically: October 6, 2008
Additional Notes: This work was supported by the Ewha Womans University Research Grant of 2007
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society