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Transactions of the American Mathematical Society

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

 

 

All infinite groups are Galois groups over any field


Authors: Manfred Dugas and Rüdiger Göbel
Journal: Trans. Amer. Math. Soc. 304 (1987), 355-384
MSC: Primary 12F10; Secondary 12F20, 20F29
DOI: https://doi.org/10.1090/S0002-9947-1987-0906820-7
MathSciNet review: 906820
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Abstract: Let $ G$ be an arbitrary monoid with $ 1$ and right cancellation, and $ K$ be a given field. We will construct extension fields $ F \supseteq K$ with endomorphism monoid End $ F$ isomorphic to $ G$ modulo Frobenius homomorphisms. If $ G$ is a group, then Aut $ F = G$. Let $ {F^G}$ denote the fixed elements of $ F$ under the action of $ G$. In the case that $ G$ is an infinite group, also $ {F^G} = K$ and $ G$ is the Galois group of $ F$ over $ K$. If $ G$ is an arbitrary group, and $ G = 1$, respectively, this answers an open problem (R. Baer 1967, E. Fried, C. U. Jensen, J. Thompson) and if $ G$ is infinite, the result is an infinite analogue of the still unsolved Hilbert-Noether conjecture inverting Galois theory. Observe that our extensions $ K \subset F$ are not algebraic. We also suggest to consider the case $ K = {\mathbf{C}}$ and $ G = \{ 1\} $.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0906820-7
Article copyright: © Copyright 1987 American Mathematical Society