[2] The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem).
One of the major results is: given a number field F, and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing CF for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism where
This was first done by Emil Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations.
Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas.
The central aims of class field theory are: to describe G in terms of certain appropriate topological objects associated to K, to describe finite abelian extensions of K in terms of open subgroups of finite index in the topological object associated to K. In particular, one wishes to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in this topological object for K. This topological object is the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields.
is (naturally isomorphic to) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity.
This derivation is purely topological group theoretical, while to establish the axioms one has to use the ring structure of the ground field.
The origins of class field theory lie in the quadratic reciprocity law proved by Gauss.
They used additional structures: in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they use elliptic curves with complex multiplication and their points of finite order.
The famous problems of David Hilbert stimulated further development, which led to the reciprocity laws, and proofs by Teiji Takagi, Philipp Furtwängler, Emil Artin, Helmut Hasse and many others.
This combined with Pontryagin duality to give a clearer if more abstract formulation of the central result, the Artin reciprocity law.
An important step was the introduction of ideles by Claude Chevalley in the 1930s to replace ideal classes, essentially clarifying and simplifying the description of abelian extensions of global fields.
Later the results were reformulated in terms of group cohomology, which became a standard way to learn class field theory for several generations of number theorists.
As the result of local contributions by Bernard Dwork, John Tate, Michiel Hazewinkel and a local and global reinterpretation by Jürgen Neukirch and also in relation to the work on explicit reciprocity formulas by many mathematicians, a very explicit and cohomology-free presentation of class field theory was established in the 1990s.
However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case.
There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence point of view.