Class Group

Let  be a number field, then each fractional ideal  of  belongs to an equivalence class  consisting of all fractional ideals  satisfying  for some nonzero element  of . The number of equivalence classes of fractional ideals of  is a finite number, known as the class number of . 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 .

REFERENCES:

Marcus, D. A. Number Fields, 3rd ed. New York: Springer-Verlag, 1996.