Class field theory

There are methods which use cohomology groups, in particular the Brauer group, and there are methods which do not use cohomology groups and are very explicit and fruitful for applications.

However, these very explicit theories could not be extended to more general number fields. General class field theory used different concepts and constructions which work over every global field.

Often, the Langlands correspondence is viewed as a nonabelian class field theory. If and when it is fully established, it would contain a certain theory of nonabelian Galois extensions of global fields. However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case. It also does not include an analog of the existence theorem in class field theory: the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence point of view.