Cyclotomic character of a finite cyclotomic extension (${\chi }_m:\mathrm{Gal}(\mathbb{Q}({\zeta }_m)/\mathbb{Q})\simeq (\mathbb{Z}/m\mathbb{Z}{)}^{\times }$)

For a primitive $m$th root of unity ${\zeta }_m$, every Galois automorphism sends ${\zeta }_m$ to a unique ${\zeta }_m^a$ with $a$ invertible modulo $m$. The assignment $\sigma \mapsto a$ is a group homomorphism, since composition multiplies the exponents, and is injective because ${\zeta }_m$ generates the cyclotomic field. The irreducibility of cyclotomic polynomials gives degree $\phi (m)$, equal to the order of the target, so the map is an isomorphism. The cyclotomic character on an absolute Galois group factors through this finite quotient and generally has a nontrivial kernel.