classgroup

Class group, in algebraic number theory, denotes the ideal class group of a number field K. It is the abelian group Cl(K) formed by the fractional ideals of the ring of integers O_K modulo the subgroup of principal ideals. The group operation is multiplication of ideals, and the identity element is the class of principal ideals. Two nonzero ideals are considered equivalent if their quotient is a principal ideal. The class group is finite for every number field, and its order h_K is called the class number.

The class group measures the failure of unique factorization in O_K. If h_K = 1, then every ideal is principal, and O_K is a principal ideal domain, which is equivalent to unique factorization of elements in O_K. For imaginary quadratic fields, the unit group of O_K is finite, while for real quadratic fields it is infinite, reflecting different arithmetic behavior.

Examples illustrate the concept. The ring of integers Z has trivial class group. The ring Z[√-5], corresponding to the field Q(√-5), has class number 2, a classic example showing non-unique factorization: 6 can be written as 2·3 or as (1+√-5)(1-√-5), with neither factorization consisting solely of principal ideals.

Computation and variants: The class group is finite and can be computed using methods based on Minkowski bounds and ideal reduction, or, in the quadratic case, via reduced binary quadratic forms of a given discriminant. The structure of Cl(K) is a finite abelian group, often expressed as a product of cyclic groups. The narrow class group Cl^+(K) refines Cl(K) for real fields by distinguishing signs of norms of totally positive principal ideals.

Applications and context: Class groups are central in algebraic number theory and class field theory, describing unramified abelian extensions of K, and they appear in arithmetic geometry and cryptography.