Hilbert class field

希尔伯特类域

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Translation evidence: Literature usage

Meaning in context

For a number field $K$, its maximal abelian extension unramified at every finite place and split at every real place.

数域 $K$ 在所有有限素位非分歧、且所有实素位完全分裂的极大阿贝尔扩张。

Example

The Hilbert class field of $\mathbb Q(i)$ is $\mathbb Q(i)$ itself, since its class number is $1$.

Usage and distinctions

Requiring real places to split selects the ordinary class field; permitting real places to become complex gives the narrow version used in some conventions.

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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 3.3.5, printed page 99 (PDF page 107, zero-based index 106), paragraph defining H_K after Proposition 3.75 and Proposition 3.76 · Checked 2026-10-15

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§3.3.5，printed page 99 (PDF page 107, zero-based index 106)，命题3.75后的 希尔伯特类域 段落及命题3.76 · Checked 2026-10-15

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