Background. I'm reading about class field theory using a combination of Lang's Algebraic Number Theory and Tate's chapter in Cassels-Frohlich.
In Lang's chapter on ideles and adeles, he shows that any generalized ideal class group is a quotient of the idele class group. He then starts to look at norm subgroups.
In Cassels-Frohlich, Tate states that the "shift from powers to norms was decisive, and is due to Hilbert".
Question. I'm still left with the feeling that the introduction of "norm subgroups" is a bit unmotivated. Could someone perhaps point me in the direction of some motivating literature or give a concrete example (in the quadratic case or something) that shows how the transition to "thinking in norms" is natural?