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Transactions of the American Mathematical Society

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

 

 

Function fields with isomorphic Galois groups


Author: Robert J. Bond
Journal: Trans. Amer. Math. Soc. 226 (1977), 291-303
MSC: Primary 12A90; Secondary 12A55
DOI: https://doi.org/10.1090/S0002-9947-1977-0441926-4
MathSciNet review: 0441926
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Abstract: Let K be a local field or a global field of characteristic p. Let $ {G_K}$ be the Galois group of the separable closure of K over K. In the local case we show that $ {G_K}$, considered as an abstract profinite group, determines the characteristic of K and the number of elements in the residue class field. In the global case we show that $ {G_K}$ determines the number of elements in the constant field of K as well as the zeta function, genus and class number of K. Let $ K'$ be another global field of characteristic p and assume we have $ \lambda :{G_K} \to {G_{K'}}$, an isomorphism of profinite groups. Then K and $ K'$ have the same constant field, zeta function, genus and class number. We also prove that the idele class groups and divisor class groups of K and $ K'$ are isomorphic. If E is a finite extension of k, the constant field of K and $ K'$, we show that the E-rational points of the Jacobian varieties of K and $ K'$ are isomorphic as $ G(E/k)$-modules. If $ K = K'$ and $ \bar K = \bar kK$ where $ \bar k$ is the algebraic closure of k, we prove that $ \lambda ({G_{\bar K}}) = {G_{\bar K}}$ and the induced automorphism of $ G(\bar K/K)$ is the identity.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0441926-4
Keywords: Local field, function field, Galois cohomology groups, zeta function, norm residue symbol, Jacobian variety
Article copyright: © Copyright 1977 American Mathematical Society