Iwasawa theory
where  is the field generated by a primitive -th root of unity. This tower of fields has a union . Then the Galois group of  over  is isomorphic with ; because the Galois group of  over  is . In order to get an interesting Galois module here, Iwasawa took the ideal class group of , and let  be its -torsion part. There are norm mappings  when , and so an inverse system. Letting  be the inverse limit, we can say that  acts on , and it is good to have a description of this action.