norm of a number-field element

数域元素的范数

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

Other names / variants: absolute field norm

Translation evidence: Literature usage

Meaning in context

For $\alpha\in K$ with $K$ a number field, $N_{K/\mathbb Q}(\alpha)$ is the determinant of multiplication by $\alpha$ on the $\mathbb Q$-vector space $K$.

数域 $K$ 中元素 $\alpha$ 的范数，是乘法映射 $x\mapsto\alpha x$ 在 $\mathbb Q$-向量空间 $K$ 上的行列式。

Example

$N_{\mathbb Q(\sqrt2)/\mathbb Q}(1+\sqrt2)=-1$.

Usage and distinctions

A field norm may be negative; it differs from a metric norm and from the positive index defining the norm of an integral ideal.

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 1.1.1, Definition 1.1 and footnote a, printed page 1 · 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§1.1.1 定义1.1 范数，printed page 1 · Checked 2026-10-14

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