ABSTRACT
This chapter studies the structure of the genus class group and its behavior under separable base change. Base change can be a powerful tool in situations where one has some control on the amount of collapsing that can occur. The idea is to tensor with a suitable splitting field, thereby replacing field extensions (which can be rather intractable) by splitting (direct products), which can be analyzed combinatorially. The chapter reviews the basic tools for working with lattices and construct the genus class group. It describes finiteness theorems and constructs examples showing that the kernel of the map on genus class groups can be infinite, even for a finite separable extension.
