Example for the Tate's theorem, Group cohomology Tate's theorem can be found in Milne's book page 81. Can someone give an easy example for that?

This is the theorem of Tate-Nakayama. It implies local class field theory with $C = \bar{K}^\times$ (see p. 105) and global class field theory with $C$ the idele class group (see p. 198).