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

Solution

Let

e = (1, \dots, 1, 2, \dots, 2) \in R^{r + s},

with

r

entries equal to one and

s

entries equal to two. Its coordinate sum is

r + 2 s = [K : Q] \neq = 0,

e \in / H

. Since the fundamental-unit logarithms form a basis of the codimension-one hyperplane

H

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

form a basis of

R^{r + s}

. This gives unique real numbers

t, λ_{i}

such that

Log (β) = t e + \sum_{i = 1}^{m} λ_{i} Log (u_{i}) .

The sum of the coordinates on the left is

lo g ∣ N_{K / Q} (β) ∣ = lo g N ((β)) = lo g N (I) .

Every unit logarithm has coordinate sum zero, so summing the right side gives

t (r + 2 s)

. Therefore

t = \frac{lo g N ( I )}{[ K : Q ]} .

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