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There are other Artin conjectures, but they are not within the scope of this wiki
True for one-dimensional representations
The Artin conjecture is true for one-dimensional representations of the Galois group. In this case, the Artin L-function reduces to the Hecke L-function.
True for two-dimensional representations
The Artin conjecture has been settled for all two-dimensional representations:
- The cyclic and dihedral case follow from Hecke's work
- Langlands did the tetrahedral case
- Tunnell did the octahedral case, extending work of Langlands
The Brauer induction theorem states that every character of a finite group occurs as a rational linear combination of characters induced from cyclic groups. This along with Galois theory, shows that all the Artin L-functions are meromorphic.
Proof in the function fields case
The Artin conjecture has been settled in the case of function fields.