Chapter1Trailer: Abelian extensions of the rationals
Though class field theory has its origins in the law of quadratic reciprocity discovered by Gauss, its proper beginning is indicated by the Kronecker-Weber theorem, first stated by Kronecker in 1853 and proved by Weber in 1886. Although one could skip this theorem and deduce it as a consequence of more general results later on, I prefer to work through it explicitly. It will serve as a trailer for the rest of the course, giving us a preview of a number of key elements:
reciprocity laws;
passage between local and global fields, using Galois theory;
group cohomology, and applications to classifying field extensions;