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The smallest prime that splits completely in an abelian number field


Author: Paul Pollack
Journal: Proc. Amer. Math. Soc. 142 (2014), 1925-1934
MSC (2010): Primary 11R44; Secondary 11L40, 11R42
DOI: https://doi.org/10.1090/S0002-9939-2014-12199-X
Published electronically: March 5, 2014
MathSciNet review: 3182011
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Abstract: Let $ K/\mathbf {Q}$ be an abelian extension and let $ D$ be the absolute value of the discriminant of $ K$. We show that for each $ \epsilon > 0$, the smallest rational prime that splits completely in $ K$ is $ O(D^{\frac 14+\epsilon })$. Here the implied constant depends only on $ \epsilon $ and the degree of $ K$. This generalizes a theorem of Elliott, who treated the case when $ K/\mathbf {Q}$ has prime conductor.


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Additional Information

Paul Pollack
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: pollack@uga.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12199-X
Received by editor(s): July 9, 2012
Published electronically: March 5, 2014
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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