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The class number of a number field is the order $h_K=|\mathrm{Cl}(K)|$ of its ideal class group.

The S-unramified power class group of a number field is finite. For $[a]\in K(S,n)$, the fractional ideal of $a$ away from $S$ is an $n$th power ${\mathfrak{a}}^n$. Sending $[a]$ to $[\mathfrak{a}]$ gives an exact sequenceThe S-unit group is finitely generated, and the localized ideal class group is a quotient of the finite ordinary ideal class group. Both end groups are finite. This is the arithmetic finiteness needed in the Kummer-theoretic proof of the weak Mordell-Weil theorem.

The ideal class group of $K=\mathbb{Q}(\sqrt{-6})$ has order two, andThis quadratic extension of $K$ is unramified at every finite prime and therefore has the degree required of the Hilbert class field.

An ideal class is an element of the ideal class group, represented by a nonzero fractional ideal. Two fractional ideals represent the same class exactly when their quotient is a principal ideal.

For a negative fundamental discriminant $D$, primitive positive definite integral binary quadratic forms of discriminant $D$, modulo proper equivalence, correspond bijectively to the ideal class group of $\mathbb{Q}(\sqrt{D})$. The displayed ideal gives one conventional choice of this correspondence. Reduced positive definite binary quadratic forms give unique class representatives with the usual boundary sign convention, making class-number computations finite.

The map from an idele to its finite-place fractional ideal is surjective, has kernel the product of the infinite multiplicative groups and finite-place unit groups, and sends diagonal elements to principal fractional ideals. It realizes the ideal class group as the corresponding quotient of the idele group.

Iwasawa theory studies arithmetic objects in infinite towers of number fields, often Zp-extensions. Passing to inverse limits turns growing ideal class groups, unit groups and Galois groups into modules over an Iwasawa algebra. Their algebraic ranks and characteristic ideals organize the growth and relate it to p-adic L-functions.

The narrow ideal class group of a number field is the group of nonzero fractional ideals modulo principal fractional ideals generated by totally positive elements. It is the ray class group for the modulus consisting of all real Archimedean places and trivial finite part. The forgetful map to the ordinary ideal class group is surjective, with kernel $\{\pm 1{\}}^{r_1}/\mathrm{sgn}({\mathcal{O}}_K^{\times })$ under the unit signature map.

The ring $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain for the absolute field norm. Approximate both rational coefficients of a quotient by nearest integers. If the errors are $u,w\in [-1/2,1/2]$, then $|u^2-2w^2|\le 1/2<1$, giving a remainder of strictly smaller absolute norm. In particular this ring is a principal ideal domain and has trivial ideal class group.

The norm-one idele group is the kernel of the idelic modulus. Its quotient by the diagonal $K^{\times }$ is compact for a number field and surjects onto the ideal class group.