Ideal class group of a number field

For a number field $K$ , let $I_K$ be the abelian group of nonzero fractional ideals of its ring of integers, under ideal multiplication. Its ideal class group is the quotient

\operatorname{Cl}(K)=I_K/\{a\mathcal O_K:a\in K^\times\}.

Thus $[I]=[J]$ precisely when $I=aJ$ for some $a\in K^\times$ , and $[I][J]=[IJ]$ . The group law exists because $\mathcal O_K$ is a Dedekind domain.

Interpretation

The identity class consists of principal fractional ideals. Every class has an integral-ideal representative, obtained by clearing denominators. Triviality means every ideal is principal; it does not say all ideals are equal.

Example

For $K=\mathbb Q$ , every fractional ideal is $a\mathbb Z$ , so the class group is trivial.

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §3, Theorem 3.20 and definition following Remark 3.21.