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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Class groups of global function fields with certain splitting behaviors of the infinite prime
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by Yoonjin Lee PDF
Proc. Amer. Math. Soc. 137 (2009), 415-424 Request permission

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 $r$).
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Additional Information
  • Yoonjin Lee
  • Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 120-750, Republic of Korea
  • MR Author ID: 689346
  • ORCID: 0000-0001-9510-3691
  • Email: yoonjinl@ewha.ac.kr
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2448559