Solution

Start with any generator

I = (β)

and use the decomposition from part (d). For each

i

, choose an integer

n_{i}

such that

- \frac{1}{2} \leq λ_{i} + n_{i} \leq \frac{1}{2},

and put

u = u_{1}^{n_{1}} \dots u_{m}^{n_{m}}, α = β u .

Since

u

is a unit,

(α) = (β) = I

. Moreover,

Log (α) = t e + h, h = \sum_{i = 1}^{m} (λ_{i} + n_{i}) Log (u_{i}) .

The vector

h

lies in the fixed compact parallelepiped

P = {\sum_{i} c_{i} Log (u_{i}) : ∣ c_{i} ∣ \leq \frac{1}{2}} .

Hence 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

2 M_{K}

For a real embedding

σ_{i}

lo g ∣ σ_{i} (α) ∣ = t + h_{i} \leq \frac{l o g N ( I )}{[ K : Q ]} + M_{K} .

For a chosen complex embedding

τ_{j}

, the weighted coordinate gives

2 lo g ∣ τ_{j} (α) ∣ = 2 t + h_{r + j} \leq 2 \frac{l o g N ( I )}{[ K : Q ]} + 2 M_{K} .

The conjugate embedding has the same modulus. Exponentiating and taking any

C > e^{M_{K}}

∣ σ (α) ∣ < CN (I)^{1/ [K : Q]}

for every embedding

σ : K \to C

, with

C

depending only on

K

Solved by gpt-5.6-sol high.

 Ancestors (11)