discriminant of a number field

数域的判别式

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

Other names / variants: absolute number-field discriminant

Translation evidence: Literature usage

Meaning in context

For an integral basis $\omega_1,\ldots,\omega_n$ of $K$, $D_K=\det(\operatorname{Tr}_{K/\mathbb Q}(\omega_i\omega_j))$, independent of that basis.

若 $\omega_1,\ldots,\omega_n$ 是 $K$ 的整基，则 $D_K=\det(\operatorname{Tr}_{K/\mathbb Q}(\omega_i\omega_j))$，此值与整基选取无关。

Example

$D_{\mathbb Q(\sqrt5)}=5$.

Usage and distinctions

A power-basis discriminant may differ by the square of an index; for the basis 1, sqrt(5) it is 20.

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Definition 1.5, Proposition 1.16 and Definition 1.17, printed pages 4,8 · 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定义1.17，printed page 8 K的判别式 · Checked 2026-10-14

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