The cardinality of the ideal class group of a number field or Dedekind domain; it counts distinct equivalence classes of nonzero fractional ideals under principal ideal equivalence and thus quantifies the failure of every ideal to be principal.

Principle

The ideal class group is the quotient of the group of fractional ideals by the subgroup of principal ideals; its order (the class number) measures the obstruction to principalness of ideals and hence to elementwise unique factorization in arithmetic rings.

Demonstration

For the ring of integers of Q(√−5), the class number is 2, witnessed by ideals that are not principal (e.g. ideals generated by 2 and 1+√−5), while Z and the ring of integers of Q(i) have class number 1 so every ideal is principal.

Misapplication

Treating the class number as the number of distinct prime ideal factors of an element, or attempting to apply the class number defined for Dedekind domains to arbitrary non-Dedekind rings without checking hypotheses.

Consequence

A class number of 1 implies every nonzero ideal is principal; finite class number gives a finite obstruction that appears explicitly in formulae such as the analytic class number formula and influences arithmetic like the failure or success of unique factorization of elements.

Reversal

A large or infinite class number indicates many inequivalent ideal classes and a significant departure from principal ideal behaviour; in extreme contrast, class number 1 is the minimal obstruction case.

Boundary

Defined for Dedekind domains and number fields (or more generally for any domain with a well-defined ideal class group); it does not directly apply to arbitrary noncommutative rings or rings lacking a useful theory of fractional ideals.

Semantic Tension

Differs from naive counts of factorization types or from invariants of the unit group (like the regulator); narrow class number and usual class number can disagree over signatures, creating nearby but distinct notions.

Synthesis

The class number is the finite cardinal invariant of a Dedekind domain that encodes how many ideal-classes fail to be principal and thereby measures the extent to which ideal-theoretic unique factorization breaks down.