ramification index of a local number-field extension

局部数域扩张的分歧指数

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

Other names / variants: local ramification index

Translation evidence: Literature usage

Meaning in context

For a finite extension $L/K$ of finite extensions of $\mathbb Q_p$, the integer $e$ determined by $\mathfrak m_K\mathcal O_L=\mathfrak m_L^e$.

对 $\mathbb Q_p$ 的有限扩张间的有限扩张 $L/K$，分歧指数是满足 $\mathfrak m_K\mathcal O_L=\mathfrak m_L^e$ 的整数 $e$。

Example

$\mathbb Q_3(\sqrt3)/\mathbb Q_3$ has ramification index $2$.

Usage and distinctions

It measures change of uniformizer order; residue degree measures the extension of residue fields.

Recommended reading in this subject · Preparation & skip guidance

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 2.3.3, paragraph defining e and equality p O_L=P^e, printed page 49 · 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§2.3.3 分歧指数 定义段落，printed page 49 · Checked 2026-10-15

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