Class Formation - The Takagi Existence Theorem

The Takagi Existence Theorem

The main remaining theorem of class field theory is the Takagi existence theorem, which states that every finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension. The classical way to prove this is to construct some extensions with small groups of norms, by first adding in lots of roots of unity, and then taking Kummer extensions. These extensions may be non-abelian (though they are extensions of abelian groups by abelian groups); however, this does not really matter, as the norm group of a non-abelian Galois extension is the same as that of its maximal abelian extension (this can be shown using what we already know about class fields). This gives enough (abelian) extensions to show that there is an abelian extension corresponding to any finite index subgroup of the idele class group.

A consequence is that the group H0(F, AF) is exactly the idele class group modulo the connected component of the identity, or equivalently the profinite completion of the idele class group. By the Artin isomorphism, this is the abelianization of the Galois group of F.

In the case of characteristic p>0, we need to use Artin-Schreier extensions as well as Kummer extensions.

For local class field theory, it is also possible to construct abelian extensions more explicitly using Lubin-Tate formal group laws. For global fields, the abelian extensions can be constructed explicitly in many cases, but a general method for constructing all abelian extensions directly (without first constructing a larger metabelian extension) is not known.