Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 20G (Number Fields)

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

Choose the real embeddings ${\sigma }_1,\dots ,{\sigma }_r$ and one embedding ${\tau }_j$ from each pair of complex-conjugate embeddings. The logarithmic embedding of number field units isIt is a group homomorphism from multiplication to addition.

If $u$ lies in its kernel, every conjugate of $u$ has modulus one. Under the Minkowski embedding, every power $u^n$ therefore lies in one fixed bounded subset. Each $u^n$ belongs to ${\mathcal{O}}_K$, whose Minkowski image is discrete, so that bounded subset contains only finitely many such lattice points. Hence $u^a=u^b$ for some $0\le a<b$, andThus $u$ is a root of unity. Conversely, every root of unity clearly has all conjugates of modulus one and lies in the kernel. This proves the kernel of the logarithmic unit embedding is precisely $\mu (K)$.

The product formula places every unit logarithm inFor a fundamental system $u_1,\dots ,u_m$, where $m=r+s-1$, the vectorsare linearly independent, form a real basis of $H$, and generate the full latticeThis is the Dirichlet unit lattice.

Letwith $r$ entries equal to one and $s$ entries equal to two. Its coordinate sum isso $e\notin H$. Since the fundamental-unit logarithms form a basis of the codimension-one hyperplane $H$, the vectorsform a basis of ${\mathbb{R}}^{r+s}$. This gives unique real numbers $t,{\lambda }_i$ such that

The sum of the coordinates on the left isEvery unit logarithm has coordinate sum zero, so summing the right side gives $t(r+2s)$. Therefore

Start with any generator $I=(\beta )$ and use the decomposition from part (d). For each $i$, choose an integer $n_i$ such thatand putSince $u$ is a unit, $(\alpha )=(\beta )=I$. Moreover,The vector $h$ lies in the fixed compact parallelepipedHence there is a constant $M_K$ such that every real coordinate of every $h\in P$ is at most $M_K$, and every complex coordinate is at most $2M_K$.

For a real embedding ${\sigma }_i$,For a chosen complex embedding ${\tau }_j$, the weighted coordinate givesThe conjugate embedding has the same modulus. Exponentiating and taking any $C>e^{M_K}$ proves the archimedean balancing of a principal ideal generator:for every embedding $\sigma :K\to \mathbb{C}$, with $C$ depending only on $K$.