Let K be a number field, then each fractional ideal I of K belongs to an equivalence class consisting of all fractional ideals J satisfying for some nonzero element of K. The number of equivalence classes of fractional ideals of K is a finite number, known as the class number of K. Multiplication of equivalence classes of fractional ideals is defined in the obvious way, i.e., by letting . It is easy to show that with this definition, the set of equivalence classes of fractional ideals form an Abelian multiplicative group, known as the class group of K.

 

Class Number, Equivalence Class, Fractional Ideal