Class field theory is a branch of algebraic number theory that studies the relationships between number fields and their abelian extensions. It provides a framework for understanding how the arithmetic of these fields can be connected through their Galois groups, which describe symmetries in the roots of polynomials. The theory primarily focuses on the construction of class fields, which are extensions of number fields that have desirable properties, such as being abelian. It connects concepts like ideal class groups and local fields to broader ideas in number theory, helping mathematicians understand the structure of solutions to polynomial equations over different fields.