residue degree of a local number-field extension

局部数域扩张的惯性指数

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

Other names / variants: local residue degree · inertia degree

Translation evidence: Literature usage

Meaning in context

For a finite extension $L/K$ of $p$-adic local fields, $f(L/K)=[\mathcal O_L/\mathfrak m_L:\mathcal O_K/\mathfrak m_K]$.

对 $p$ 进局部域的有限扩张 $L/K$，惯性指数是剩余域扩张的次数 $[\mathcal O_L/\mathfrak m_L:\mathcal O_K/\mathfrak m_K]$。

Example

$\mathbb Q_3(i)/\mathbb Q_3$ has residue degree $2$, since $X^2+1$ is irreducible over $\mathbb F_3$.

Usage and distinctions

The inspected Chinese author calls f the inertia index (惯性指数); it is not the order of the inertia group, which is e in a Galois local extension.

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, f=[kappa_L:kappa] and Proposition 2.39, 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 明称剩余域扩张次数为 惯性指数，printed page 49 · Checked 2026-10-15

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