ideal class group of a number field

数域的理想类群

Number Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: number-field class group

Translation evidence: Literature usage

Meaning in context

The quotient of the multiplicative group of nonzero fractional $\mathcal O_K$-ideals by the subgroup of principal fractional ideals.

非零分式 $\mathcal O_K$-理想的乘法群对主分式理想子群的商群。

Example

$\operatorname{Cl}(\mathbb Q(i))$ is trivial since $\mathbb Z[i]$ is a principal ideal domain.

Usage and distinctions

It measures failure of ideals to be principal; it is different from the group of units of the integer ring.

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Definition sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义Section 1.3.2, quotient Cl_K=I_K/P_K, printed page 15 · Checked 2026-10-14

Translation sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义§1.3.2，printed page 15 理想类群 · Checked 2026-10-14

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