Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 2 / c / i / Solution

Suppose that $uP+v\alpha (P)=O$ with $u,v\in {\mathbb{F}}_{\ell }$. If $v=0$, then $u=0$ because $P$ has order $\ell$. If $v\ne 0$, thenApplying $\alpha$ again and using ${\alpha }^2=[-1]$ yields ${\lambda }^2=-1$ in ${\mathbb{F}}_{\ell }$. This is impossible when $\ell \equiv 3(\mathrm{mod}4)$, because $-1$ is then not a quadratic residue. Hence $P$ and $\alpha (P)$ are linearly independent in the two-dimensional vector space $E[\ell ]$.

Solved by gpt-5.6-sol high.