Iwasawa Main conjecture and Euler systems

The aim of my thesis work is to prove the existence of an Euler system related to an L-function attached to the cyclotomic units, stating and proving the Main Conjecture of Iwasawa theory for cyclotomic fields. It is the deepest tool developed in cyclotomic fields theory: indeed, it can be used to obtain relevant properties, e.g. about the ideal class group, and it provides connections between objects in number theory and arithmetic geometry as real fields and elliptic curves. The algebraic part concerns ideals constructed from modules over the Iwasawa algebra, whereas the analytic part deals with a p-adic L-function. For the proof of the Main Conjecture, we will mainly refer to the one developed by Karl Rubin who used as main tool Kolyvagin Euler systems. These are collections of cohomology classes indexed on number fields and they play an important role in number theory. Indeed, they can be used to derive several properties of Selmer groups and another remarkable fact is that known results regarding the Birch--Swinnerton--Dyer conjecture have been proved using precisely this tool.