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

Lee, Geon-No; Kwon, Soun-Hi

doi:10.1090/S0025-5718-08-02093-0

The zeros of Dedekind zeta functions and class numbers of CM-fields
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by Geon-No Lee and Soun-Hi Kwon;

Math. Comp. 77 (2008), 2437-2445

DOI: https://doi.org/10.1090/S0025-5718-08-02093-0

Published electronically: June 2, 2008

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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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