Class field theory
![The fifth roots of unity in the complex plane. Adjoining these roots to the rational numbers generates an abelian extension.](/Images/godic/202501/26/One5Root.svg4410.png")
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields and function fields of curves over finite fields and arithmetic properties of such abelian extensions. A general name for such fields is global fields, or one-dimensional global fields.