Iwasawa theory
The motivation here was undoubtedly that the -torsion in the ideal class group of   had already been identified by Kummer as the main obstacle to the direct proof of Fermat's last theorem. What Iwasawa did that was new, was to go 'off to infinity' in a new direction. In fact,   is a module over the group ring . This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.