quadratic number field

二次数域

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

Other names / variants: quadratic field

Translation evidence: Literature usage

Meaning in context

A degree-two extension of Q\mathbb Q, equivalently Q(d)\mathbb Q(\sqrt d) for a squarefree integer d≠0,1d\ne0,1.

有理数域的二次扩张；等价地，形如 Q(d)\mathbb Q(\sqrt d)，其中 d≠0,1d\ne0,1 为无平方因子整数。

Example

Q(−3)\mathbb Q(\sqrt{-3}) is an imaginary quadratic field.

Usage and distinctions

The sign of d separates real and imaginary quadratic fields; adjoining a square root to an arbitrary base field is a broader construction.

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 1.2.2, Example 1.2 and Section 1.2.3, Exercise 1.2.8(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§1.2.3，printed page 10 习题1.2.8(1) 二次域；§1.2.2 例题1.2 · Checked 2026-10-14
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