 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-166/3/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

For

β \in K

γ = α - β

has degree at most

[Q (α, β) : Q] \leq 3 \cdot 2 = 6.

The basic height inequality gives

H (γ) \leq 2 H (α) H (β) .

∣ α - β ∣ \geq H (γ)^{- 6} \geq (2 H (α))^{- 6} H (β)^{- 6} .

Thus one may take the explicit constant

c = (2 H (α))^{- 6} .

 New to topics? Read the docs here!