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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
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