Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 20G / a

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- Solution a

Solution

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 .

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