trace of a number-field element

数域元素的迹

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

Other names / variants: absolute field trace

Translation evidence: Literature usage

Meaning in context

For a number field $K$ and $\alpha\in K$, $\operatorname{Tr}_{K/\mathbb Q}(\alpha)$ is the trace of the $\mathbb Q$-linear map $x\mapsto\alpha x$ on $K$.

数域 $K$ 中元素 $\alpha$ 的迹，是乘法映射 $x\mapsto\alpha x$ 作为 $K$ 上 $\mathbb Q$-线性变换的迹。

Example

$\operatorname{Tr}_{\mathbb Q(\sqrt2)/\mathbb Q}(3+\sqrt2)=6$.

Usage and distinctions

The ambient extension matters: the trace of 3 from a quadratic field is 6, although its minimal polynomial over Q has degree one.

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, 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 迹，printed page 1 · Checked 2026-10-14

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