Take K a number field and L an Abelian extension, then form a prime divisor m that is divided by all ramified primes of the extension L/K. Now define a map ϕ_(L/K) from the fractional ideals relatively prime to m to the Galois group of L/K that sends an ideal a to ((L/K), a). This map is called the Artin map. Its importance lies in the kernel, which Artin's reciprocity theorem states contains all fractional ideals that are only composed of primes that split completely in the extension L/K.

Artin symbol | class field | class field theory | Hilbert class field