 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/20g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

K

have

r

real embeddings and

s

conjugate pairs of complex embeddings. The Dirichlet unit theorem states that

O_{K}^{\times} ≅ μ (K) \times Z^{r + s - 1},

where

μ (K)

is the finite cyclic group of roots of unity in

K

. Thus the unit rank is

m = r + s - 1.

Equivalently, there are units

u_{1}, \dots, u_{m}

such that every unit has a unique expression

u = ζ u_{1}^{n_{1}} \dots u_{m}^{n_{m}}, ζ \in μ (K), n_{i} \in Z .

 New to topics? Read the docs here!