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Dedekind Domain
Linked via "class number"
where $\mathcal{I}(R)$ is the set of non-zero fractional ideals of $R$, and $\mathcal{P}(R)$ is the set of non-zero principal fractional ideals.
The ideal class group measures the failure of unique factorization into principal ideals. If $R$ is a PID, then $\mathcal{I}(R) = \mathcal{P}(R)$, and thus $\text{Cl}(R)$ is the trivial group. The order of the class group, $|\text{Cl}(R)|$, is known as the class number.
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Dedekind Domain
Linked via "class number"
Example: The Field $\mathbb{Q}(\sqrt{-19})$
The ring of integers $\mathcal{O}_K$ for $K = \mathbb{Q}(\sqrt{-19})$ is $R = \mathbb{Z}[\omega]$ where $\omega = \frac{1+\sqrt{-19}}{2}$. This domain is known to be a PID, and thus its class number is 1. The apparent counterexample factorizations sometimes cited, such as the one involving seven factors for the number 7 (see Table 1), actually represent the factorization of the principal ideal $(7)$ into prime ideals, not the factori… -
Discriminant
Linked via "class number"
$$d(K) = \left( \det \begin{pmatrix} \sigma1(\omega1) & \dots & \sigma1(\omegan) \\ \vdots & \ddots & \vdots \\ \sigman(\omega1) & \dots & \sigman(\omegan) \end{pmatrix} \right)^2$$
The sign and parity of $d(K)$ carry deep significance. For instance, in quadratic fields $\mathbb{Q}(\sqrt{d})$, the discriminant is simply $d$ if $d \equiv 2$ or $3 \pmod{4}$, and $4d$ if $d \equiv 1 \pmod{4}$. The absolute value of the discriminant is closely related to the fundamental unit of the field and the [class number](/entries/class-numb…