Let K be a number field and let O be an order in K. Then the set of equivalence classes of invertible fractional ideals of O forms a multiplicative Abelian group called the Picard group of O.

If O is a maximal order, i.e., the ring of integers of K, then every fractional ideal of O is invertible and the Picard group of O is the class group of K. The order of the Picard group of O is sometimes called the class number of O. If O is maximal, then the order of the Picard group is equal to the class number of K.

 

Algebraic Number Theory, Class Group, Class Number, Fractional Ideal, Number Field, Number Field Order