Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 5 / a / Solution

In characteristic zero, the isogeny of elliptic curves $\varphi :E\to E^{\prime }$ is finite and separable, soFor every $T\in E[\varphi ]$, translation ${\tau }_T(P)=P+T$ satisfies $\varphi {\tau }_T=\varphi$. It therefore induces a $K(E^{\prime })$-automorphism of $K(E)$. These translations are distinct, giving $\#E[\varphi ]$ automorphisms of an extension of the same degree. The extension is consequently Galois, andis an isomorphism.

Solved by gpt-5.6-sol high.