What is a Dedekind domain, what is its ideal class group, and what is the class number?
Solution
A Dedekind domain is an integral domain \(R\) satisfying any of the standard equivalent conditions; one common definition is:
- \(R\) is Noetherian;
- \(R\) is integrally closed in its fraction field;
- every nonzero prime ideal is maximal.
Equivalently, every nonzero proper ideal factors uniquely as a product of nonzero prime ideals.
Let \(K=\operatorname{Frac}(R)\). The nonzero fractional ideals of \(R\) form an abelian group under multiplication. The nonzero principal fractional ideals \[ (a)=aR, \qquad a\in K^\times, \] form a subgroup. The quotient \[ \operatorname{Cl}(R) =\{\text{nonzero fractional ideals}\}/\{\text{principal fractional ideals}\} \] is the ideal class group.
Its elements measure the obstruction to ideals being principal. In particular, \[ \operatorname{Cl}(R)=0 \iff R\text{ is a PID}. \]
When the class group is finite, its order \[ h(R)=|\operatorname{Cl}(R)| \] is the class number. For the ring of integers \(\OO_K\) of a number field, the class group is finite, so the class number is always defined. Class number \(1\) means that \(\OO_K\) is a PID and hence a UFD.