Dirichlet's unit theorem

狄利克雷单位定理

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

Translation evidence: Literature usage

Meaning in context

If a number field has $r_1$ real embeddings and $r_2$ conjugate pairs of complex embeddings, then $\mathcal O_K^\times\cong\mu_K\times\mathbb Z^{r_1+r_2-1}$, where $\mu_K$ is its group of roots of unity.

若数域有 $r_1$ 个实嵌入和 $r_2$ 对共轭复嵌入，则 $\mathcal O_K^\times\cong\mu_K\times\mathbb Z^{r_1+r_2-1}$，其中 $\mu_K$ 为单位根群。

Example

For $\mathbb Q(\sqrt2)$ the unit rank is $2+0-1=1$.

Usage and distinctions

The rank uses $r_1+r_2$, whereas the field degree is $r_1+2r_2$; roots of unity are the finite torsion part.

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 1.3.2, Theorem 1.37, printed page 15 · 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理1.37 狄利克雷单位定理，printed page 15 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.

Open the reading guide for this concept