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

Let $K$ have $r$ real embeddings and $s$ conjugate pairs of complex embeddings. The Dirichlet unit theorem states thatwhere $\mu (K)$ is the finite cyclic group of roots of unity in $K$. Thus the unit rank isEquivalently, there are units $u_1,\dots ,u_m$ such that every unit has a unique expression