Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 5 / a / ii / Solution

Take $f=g$ and $L=X_1+\cdots +X_n$. Permuting the variables produces another solution with the same linear term, so uniqueness givesFor $a\in {\mathcal{O}}_K$, the one-variable case of part i gives a unique ${\theta }_a(X)\equiv aX(\mathrm{mod}{X}^2)$ commuting with $f$. Applying uniqueness once more to the two ways of composing ${\theta }_a$ with $F$ givesThus $F$ is the addition law and the ${\theta }_a$ are scalar endomorphisms of the Lubin–Tate formal group.