Example
Let p be a prime number and let K = Q(μp) be the field generated over Q by the pth roots of unity. Iwasawa considered the following tower of number fields:
where is the field generated by adjoining to the pn+1st roots of unity and . The fact that implies, by infinite Galois theory, that is isomorphic to . In order to get an interesting Galois module here, Iwasawa took the ideal class group of, and let be its p-torsion part. There are norm maps whenever, and this gives us the data of an inverse system. If we set, then it is not hard to see from the inverse limit construction that is a module over . In fact, is a module over the Iwasawa algebra (i.e. the completed group ring of over ). This is a well-behaved ring (a 2-dimensional, regular local ring), and this makes it possible to classify modules over it in a way that is not too coarse. From this classification it is possible to recover information about the p-part of the class group of .
The motivation here is that the p-torsion in the ideal class group of had already been identified by Kummer as the main obstruction to the direct proof of Fermat's last theorem.
Read more about this topic: Iwasawa Theory
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