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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Radical and cyclotomic extensions of the rational numbers


Authors: David Gluck and I. M. Isaacs
Journal: Proc. Amer. Math. Soc. 135 (2007), 3435-3441
MSC (2000): Primary 12F10
DOI: https://doi.org/10.1090/S0002-9939-07-08864-8
Published electronically: August 1, 2007
MathSciNet review: 2336555
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Abstract: A radical extension of the rational numbers $ \mathbb{Q}$ is a field $ R \supseteq \mathbb{Q}$ generated by an element having a power in $ \mathbb{Q}$, and a cyclotomic extension $ K \supseteq \mathbb{Q}$ is an extension generated by a root of unity. We show that a radical extension that is almost Galois over $ \mathbb{Q}$ is almost cyclotomic. More precisely, we prove that if $ R$ is radical with Galois closure $ E$, then $ E$ contains a cyclotomic field $ K$ such that the degree $ \vert E:K\vert$ is bounded above by an almost linear function of $ \vert E:R\vert$. In particular, if $ R$ is Galois, it contains a cyclotomic field $ K$ such that $ \vert R:K\vert \le 3$.


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Additional Information

David Gluck
Affiliation: Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, Michigan 48202
Email: dgluck@math.wayne.edu

I. M. Isaacs
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: isaacs@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08864-8
Received by editor(s): July 5, 2006
Published electronically: August 1, 2007
Communicated by: Martin Lorenz
Article copyright: © Copyright 2007 American Mathematical Society