= Cyclotomic character of a finite cyclotomic extension
{c}
{title2=$\chi_m:\operatorname{Gal}(\mathbb Q(\zeta_m)/\mathbb Q)\simeq(\mathbb Z/m\mathbb Z)^\times$}
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 $\varphi(m)$, 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>.
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