Picard Group

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

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