= Cyclotomic Artin reciprocity over the rational numbers
{title2=$\operatorname{Art}((a)):\zeta_n\mapsto\zeta_n^a$}
For the <cyclotomic field> $\mathbb Q(\zeta_n)$, arithmetic <Frobenius automorphisms> at primes $\ell\nmid n$ send $\zeta_n$ to $\zeta_n^\ell$: reduction distinguishes the nth <roots of unity>, and <Frobenius> raises residues to their $\ell$th powers. Multiplicativity therefore sends a <fractional ideal> with positive rational generator $a$, prime to $n$, to the exponent $a\bmod n$. Its <kernel> consists exactly of positive generators congruent to one modulo $n$. This proves <Artin reciprocity> for the <ray class group> of modulus $n\infty$ without assuming the general reciprocity theorem.
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