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The zeros of Dedekind zeta functions and class numbers of CM-fields


Authors: Geon-No Lee and Soun-Hi Kwon
Journal: Math. Comp. 77 (2008), 2437-2445
MSC (2000): Primary 11R29, 11R42
DOI: https://doi.org/10.1090/S0025-5718-08-02093-0
Published electronically: June 2, 2008
MathSciNet review: 2429892
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Abstract: Let $ F'/F$ be a finite normal extension of number fields with Galois group $ Gal(F'/F)$. Let $ \chi$ be an irreducible character of $ Gal(F'/F)$ of degree greater than one and $ L(s,\chi)$ the associated Artin $ L$-function. Assuming the truth of Artin's conjecture, we have explicitly determined a zero-free region about $ 1$ for $ L(s,\chi)$. As an application we show that, for a CM-field $ K$ of degree $ 2n$ with solvable normal closure over $ \mathbb{Q}$, if $ n \geq 370$ as well as $ n \notin \{ 384, 400, 416, 448, 512 \}$, then the relative class number of $ K$ is greater than one.


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Additional Information

Geon-No Lee
Affiliation: Department of Mathematics, Korea University, 136-701, Seoul, Korea
Email: thisknow@korea.ac.kr

Soun-Hi Kwon
Affiliation: Department of Mathematics Education, Korea University, 136-701, Seoul, Korea
Email: sounhikwon@korea.ac.kr

DOI: https://doi.org/10.1090/S0025-5718-08-02093-0
Keywords: CM-fields, class numbers, relative class numbers, Dedekind zeta functions
Received by editor(s): July 6, 2018
Received by editor(s) in revised form: August 22, 2007
Published electronically: June 2, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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