The Artin reciprocity says that if $L/\mathbb{Q}$ is a finite abelian extension with defining modulus $m$, then the sequence of groups
$$ 1\to I_{L,m}\to (\mathbb{Z}/m\mathbb{Z})^\times\to Gal(L/\mathbb{Q})\to 1 $$
is exact.
I do not quite see why this is a substantial theorem, because it is simply saying that the Artin map (the third arrow) is surjective; $I_{L,m}$ is defined as the kernel of the Artin map, so the second arrow being injective is trivial. Also, why is it called "reciprocity"? Does this theorem have any interesting consequences?