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Transactions of the American Mathematical Society

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



Galois groups and the multiplicative structure of field extensions

Authors: Robert Guralnick and Roger Wiegand
Journal: Trans. Amer. Math. Soc. 331 (1992), 563-584
MSC: Primary 12F05; Secondary 12F10
MathSciNet review: 1036008
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Abstract: Let $ K/k$ be a finite Galois field extension, and assume $ k$ is not an algebraic extension of a finite field. Let $ {K^{\ast} }$ be the multiplicative group of $ K$, and let $ \Theta (K/k)$ be the product of the multiplicative groups of the proper intermediate fields. The condition that the quotient group $ \Gamma = {K^{\ast} }/\Theta (K/k)$ be torsion is shown to depend only on the Galois group $ G$. For algebraic number fields and function fields, we give a complete classification of those $ G$ for which $ \Gamma $ is nontrivial.

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Keywords: Multiplicative group of a field, Galois group, representation, character, finite simple group
Article copyright: © Copyright 1992 American Mathematical Society