ideal class (original) (raw)

The number of equivalence classes, denoted by h or hK, is called the class number of K.

Note that the set of ideals of any ring R forms an abelian semigroup with the product of ideals as the semigroup operation. By replacing ideals by ideal classes, it is possible to define a group on the ideal classes of 𝒪K in the following way.

Let 𝔞, 𝔟 be ideals of 𝒪K. Denote the ideal classes of which 𝔞 and 𝔟 are representatives by [𝔞] and [𝔟] respectively. Then define ⋅ by

Let 𝒞={[𝔞]∣𝔞≠(0),𝔞⁢ an ideal of ⁢𝒪K}. With the above definition of multiplication, 𝒞 is an abelian group, called the ideal class group (or frequently just the class group) of K.