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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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On Hilbert class fields in characteristic $p>0$ and their $L$-functions
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by Stuart Turner PDF
Proc. Amer. Math. Soc. 64 (1977), 39-42 Request permission

Abstract:

Let k be a global field of characteristic $p > 0$ with field of constants ${{\mathbf {F}}_q}$. Let $\bar k$ be an algebraic closure of k. In this note we study the subfields of $\bar k$ which are maximal unramified abelian extensions of k with field of constants ${{\mathbf {F}}_q}$. Each of these fields may be regarded as an analogue of the Hilbert class field of algebraic number theory [1, p. 79]. In §1 we recall the construction of these class fields and in §2 we show that if k has genus one, they are all ${{\mathbf {F}}_q}$-isomorphic. In §3 we show that this is not necessarily the case if the genus of k is greater than one. The argument there is based on an observation about the L-functions of the fields.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 39-42
  • MSC: Primary 12A65
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0439813-6
  • MathSciNet review: 0439813