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A one-dimensional Artin representation $\rho:\Gal(\overline{\Q}/\Q)\to \GL_1(\C)\cong \C^\times$. Since the image is abelian and every Artin representation factors through the Galois group of a number field, we can view $\rho$ as a map $\Gal(K/\Q)\to \C^\times$ for an abelian number field $K$.

By the Kronecker-Weber theorem, $K$ is contained in a cyclotomic extension $\Q(\zeta_n)$ for some positive integer $n$. Using the canonical isomorphism $\Gal(\Q(\zeta_n)/\Q)\cong(\Z/n\Z)^\times$, the Artin representation corresponds to a homomorphism $(\Z/n\Z)^\times\to \C^\times$.

Extending it to $\Z/n\Z$ by defining it to be $0$ off of $(\Z/n\Z)^\times$, and then viewing it as a function on $\Z$ via the natural map $\Z\to\Z/n\Z$, we get the corresponding Dirichlet character, whose modulus is the same as that of the one-dimensional Artin representation.

Knowl status:
  • Review status: reviewed
  • Last edited by Kiran S. Kedlaya on 2019-05-02 23:52:13
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