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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
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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Article copyright: © Copyright 1977 American Mathematical Society