fundamental unit of a real quadratic field

实二次域的基本单位

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Other names / variants: fundamental unit in a real quadratic field

Translation evidence: Literature usage

Meaning in context

For a real quadratic field $K$, a unit $\varepsilon$ generating $\mathcal O_K^\times/\{\pm1\}$, so every unit is $\pm\varepsilon^n$ for some $n\in\mathbb Z$.

对实二次域 $K$，基本单位 $\varepsilon$ 是 $\mathcal O_K^\times/\{\pm1\}$ 的生成元；每个单位均可写成 $\pm\varepsilon^n$。

Example

$1+\sqrt2$ is a fundamental unit of $\mathbb Q(\sqrt2)$.

Usage and distinctions

A fundamental unit need not have norm +1; squaring a norm -1 generator produces a smaller norm-one subgroup.

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 1.3.4, opening unit-group description, printed page 17 · 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§1.3.4 开头 基本单位，printed page 17 · Checked 2026-10-14

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