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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
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.

References [Enhancements On Off] (What's this?)

  • [1] E. Artin and J. Tate, Class field theory, Benjamin, New York, 1968. MR 36 #6383. MR 0223335 (36:6383)
  • [2] J. Neukirch, Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper, Invent. Math. 6 (1969), 296-314. MR 39 #5528. MR 0244211 (39:5528)
  • [3] D. S. Rim and G. Whaples, Global norm-residue map over quasi-finite field, Nagoya Math. J. 27 (1966), 323-329. MR 34 #4252. MR 0204410 (34:4252)
  • [4] J. P. Serre, Corps locaux, 2nd ed., Hermann, Paris, 1968. MR 0354618 (50:7096)
  • [5] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144. MR 34 #3749. MR 0206004 (34:5829)

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Keywords: Local field, function field, Galois cohomology groups, zeta function, norm residue symbol, Jacobian variety
Article copyright: © Copyright 1977 American Mathematical Society

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