The smallest prime that splits completely in an abelian number field
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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 $\varepsilon > 0$, the smallest rational prime that splits completely in $K$ is $O(D^{\frac 14+\varepsilon })$. Here the implied constant depends only on $\varepsilon$ and the degree of $K$. This generalizes a theorem of Elliott, who treated the case when $K/\mathbf {Q}$ has prime conductor.References
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Additional Information
- Paul Pollack
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 830585
- Email: pollack@uga.edu
- Received by editor(s): July 9, 2012
- Published electronically: March 5, 2014
- Communicated by: Ken Ono
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3182011