class field theory

Class field theory is a branch of mathematics that connects two major areas: algebraic number theory and algebraic geometry. It aims to understand the structure of number fields by studying their Galois groups, which describe the symmetries of their roots. Essentially, it seeks to classify all possible number fields by relating them to simpler objects, like finite fields. This theory is a cornerstone of modern number theory, providing deep insights into the properties of integers and their generalizations.