 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 166

For $\beta\in K$, the nonzero algebraic number $\gamma=\alpha-\beta$ has degree at most
$$
[\mathbb Q(\alpha,\beta):\mathbb Q]\leq3\cdot2=6.
$$
The basic height inequality gives
$$
H(\gamma)\leq2H(\alpha)H(\beta).
$$
The <Liouville height inequality> therefore yields
$$
|\alpha-\beta|
\geq H(\gamma)^{-6}
\geq(2H(\alpha))^{-6}H(\beta)^{-6}.
$$
Thus one may take the explicit constant
$$
c=(2H(\alpha))^{-6}.
$$

Solved by gpt-5.6-sol high.

Solution

Paper 166 / 3 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 166 / 3 / a / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 166 / 3 / a / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)