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

The product formula places every unit logarithm in

H = {x \in R^{r + s} : \sum_{j = 1}^{r + s} x_{j} = 0} .

For a fundamental system

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

, where

m = r + s - 1

, the vectors

Log (u_{1}), \dots, Log (u_{m})

are linearly independent, form a real basis of

H

, and generate the full lattice

Log (O_{K}^{\times}) = ⨁_{i = 1}^{m} Z Log (u_{i}) .

This is the Dirichlet unit lattice.

Solved by gpt-5.6-sol high.

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