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Proceedings of the American Mathematical Society

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

 

 

On Hilbert class fields in characteristic $ p>0$ and their $ L$-functions


Author: Stuart Turner
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
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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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DOI: https://doi.org/10.1090/S0002-9939-1977-0439813-6
Article copyright: © Copyright 1977 American Mathematical Society