Let K be a number field, then each fractional ideal I of K belongs to an equivalence class [I] consisting of all fractional ideals J satisfying I = α J for some nonzero element α 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 [I][J] = [I J]. 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.
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