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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Class numbers of cyclotomic function fields
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by Li Guo and Linghsueh Shu PDF
Trans. Amer. Math. Soc. 351 (1999), 4445-4467 Request permission

Abstract:

Let $q$ be a prime power and let ${\mathbb F}_q$ be the finite field with $q$ elements. For each polynomial $Q(T)$ in ${\mathbb F}_q [T]$, one could use the Carlitz module to construct an abelian extension of ${\mathbb F}_q (T)$, called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of ${\mathbb F}_q(T)$, similar to the role played by cyclotomic number fields for abelian extensions of ${\mathbb Q}$. We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in ${\mathbb F}_q [T]$. Two types of properties are obtained for the $l$-parts of the class numbers of the fields in this tower, for a fixed prime number $l$. One gives congruence relations between the $l$-parts of these class numbers. The other gives lower bound for the $l$-parts of these class numbers.
References
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Additional Information
  • Li Guo
  • Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
  • Address at time of publication: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102
  • Email: liguo@andromeda.rutgers.edu
  • Linghsueh Shu
  • Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
  • Address at time of publication: 228 Paseo del Rio, Moraga, California 94556
  • Email: shul@wellsfargo.com
  • Received by editor(s): May 15, 1997
  • Published electronically: June 10, 1999
  • Additional Notes: The authors were supported in part by NSF Grants #DMS-9301098 and #DMS-9525833.
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4445-4467
  • MSC (1991): Primary 11R29, 11R58; Secondary 11R23
  • DOI: https://doi.org/10.1090/S0002-9947-99-02325-9
  • MathSciNet review: 1608317