Class field theory: the century-long project that grew from Gauss's reciprocity law
David Hilbert usually gets credit for class fields, but Leopold Kronecker already knew the idea and Heinrich Weber coined the name before Hilbert's key papers appeared. What followed was a relay across several generations of number theorists, finishing most of its central results by 1940 and producing a blueprint for the ambitious Langlands program.
The theory's aim is to catalogue a particular family of number systems, the abelian extensions of a field, whose symmetry groups are especially orderly, using only information found inside the starting field. Its roots lie in Gauss's quadratic reciprocity law, which predicts when one prime is a perfect square in the arithmetic of another, and it grew through Ernst Kummer's work on ideals, Kurt Hensel's completions and research on roots of unity.
For the ordinary rational numbers there is a strikingly concrete answer. The Kronecker–Weber theorem, first conjectured by Kronecker, says every such extension of the rationals can be built from roots of unity, the complex solutions of equations like x to the n equals 1. For imaginary quadratic fields, elliptic curves with a special symmetry called complex multiplication play the same role. Neither recipe carries over to number fields in general, which forced a more abstract approach.
Hilbert's famous problems pushed the work forward, and Teiji Takagi, Philipp Furtwängler, Emil Artin and Helmut Hasse supplied proofs. Takagi's existence theorem was in hand by 1920, and Artin's reciprocity law became the heart of the subject. Early proofs leaned heavily on analysis. In the 1930s Claude Chevalley introduced objects called ideles that simplified the whole description, and Wolfgang Krull's work on infinite extensions gave a cleaner formulation.
Artin and John Tate later recast everything through group cohomology, the standard route for generations of students, though criticised for being hard to compute with. By the 1990s Jürgen Neukirch, building on work by Bernard Dwork, Tate and Michiel Hazewinkel, had produced an explicit version free of cohomology. The Langlands correspondence is often described as the nonabelian sequel, though it lacks the tidy existence theorem that made class fields so useful.
Source: Class field theory
