Maximal abelian extension
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields (one-dimensional local fields) and "global fields" (one-dimensional global fields) such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields. It also studies various arithmetic properties of such abelian extensions. Class field theory includes global class field theory and local class field theory.
The abelian topological group CK associated to such a field K is the multiplicative group of a local field or the idele class group of a global field.
One of fundamental results of class field theory is a construction of a nontrivial reciprocity homomorphism, which acts from CK to the Galois group of the maximal abelian extension of the field K. The existence theorem of class field theory states that each open subgroup of finite index of CK is the image with respect to the norm map from the corresponding class field extension down to K.
The theory takes its name from the fact that it includes a one-to-one correspondence between finite abelian extensions of a fixed local or global field and appropriate open subgroup of finite index in CK. For example, in the case of number fields, the latter are classes of ideals of the field or open subgroups of the idele class group of the field; the Hilbert class field, which is the maximal unramified abelian extension of a number field, corresponds to a very special class of ideals.
A standard method since the 1930s is to develop local class field theory, which describes abelian extensions of completions of a local field, and then use it to construct global class field theory.
...
Wikipedia