Picard Group

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.

See also

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

This entry contributed by David Terr

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Terr, David. "Picard Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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