Nonmonomial characters and Artin’s conjecture

Foote, Richard

doi:10.1090/S0002-9947-1990-0987161-9

Nonmonomial characters and Artin’s conjecture
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Trans. Amer. Math. Soc. 321 (1990), 261-272 Request permission

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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