北京雁栖湖应用数学研究院

The goal of class field theory is to describe the Galois extensions of a local or global field in terms of the arithmetic of the field itself. For abelian extensions, the theory was developed between roughly 1850 and 1930 by Kronecker, Weber, Hilbert, Takagi, Artin, Hasse, and others. This course begins with an analysis of the quadratic case of Class Field Theory via Hilbert symbols, in order to give a more hands-on introduction to the ideas of Class Field Theory. More advanced topics in number theory are discussed in this course, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, and quadratic forms.