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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nonmonomial characters and Artin’s conjecture


Author: Richard Foote
Journal: Trans. Amer. Math. Soc. 321 (1990), 261-272
MSC: Primary 11R42; Secondary 11R32
DOI: https://doi.org/10.1090/S0002-9947-1990-0987161-9
MathSciNet review: 987161
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Abstract: If $E/F$ is a Galois extension of number fields with solvable Galois group $G$, the main result of this paper proves that if the Dedekind zeta-function of $E$ has a zero of order less than ${\mathcal {M}_G}$ at the complex point ${s_0} \ne 1$, then all Artin $L$-series for $G$ are holomorphic at ${s_0}$ — here ${\mathcal {M}_G}$ is the smallest degree of a nonmonomial character of any subgroup of $G$. The proof relies only on certain properties of $L$-functions which are axiomatized to give a purely character-theoretic statement of this result.


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Article copyright: © Copyright 1990 American Mathematical Society