Dirichlet character

狄利克雷特征

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Other names / variants: Dirichlet multiplicative character

Translation evidence: Literature usage

Meaning in context

For $N\ge1$, a homomorphism $(\mathbb Z/N\mathbb Z)^\times\to\mathbb C^\times$, extended periodically to integers and assigned $0$ on integers not coprime to $N$.

对正整数 $N$，从 $(\mathbb Z/N\mathbb Z)^\times$ 到 $\mathbb C^\times$ 的群同态；周期性延拓到整数，并在与 $N$ 不互素的整数上取零。

Example

Modulo $4$, $\chi(n)=0$ for even $n$, $1$ for $n\equiv1$, and $-1$ for $n\equiv3$.

Usage and distinctions

Its restriction to units is a group character; the extension to all integers has zeros and is not a group homomorphism on their additive group.

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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 4.1.1, Definition 4.2, printed page 103 · 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4.2 狄利克雷特征，printed page 103 · Checked 2026-10-14

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