Galois Groups Acting on Ideal Class Groups

Relatively recently, it has been observed, in particular by Iwasawa and Leopoldt, that the action of Galois groups on ideal class groups can be used to great advantage to reinterpret old results and to obtain new information on the structure of class groups. In this chapter we first give some results which are useful when working with class groups and class numbers. We then present the basic machinery, essentially Leopoldt’s Spiegelungssatz, which underlies the rest of the chapter. As applications, Kummer’s result “ $$ p\left| {{h^ + } \Rightarrow p} \right|{h^ - }$$ ” is made more precise and a classical result of Scholz on class groups of quadratic fields is proved. Finally, we show that Vandiver’s conjecture implies that the ideal class group of $$ \mathbb{Q}\left( {{\zeta _{{p^n}}}} \right)$$is isomorphic to the minus part of the group ring modulo the Stickelberger ideal.