We know that the failure the unique factorisation is measured by the ideal class-group, that of the local-global principle depends upon the Tate-Shafarevich group.
Then I thought: what should be the right tool to detect the failure of the Hasse-norm theorem?
Also, when does the Hasse-norm theorem hold, apart from the cyclic cases?
I know almost nothing in this direction, so any reference or indication are the most appreciated; thanks in advance.