In number theory , Iwasawa theory is a Galois module theory of ideal class groups , started by Kenkichi Iwasawa , in the 1950s , as part of the theory of cyclotomic fields . In number theory , Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields . It began as a Galois module theory of ideal class groups , initiated by Kenkichi Iwasawa , in the 1950s , as part of the theory of cyclotomic fields .