Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 1 / a / Solution

A one-dimensional commutative formal group law over $R$ is a power series $F(X,Y)\in R[[X,Y]]$ satisfyingA homomorphism $\theta :\mathcal{F}\to \mathcal{G}$ is a series $\theta (T)\in TR[[T]]$ such that

Write $\theta (T)=uT+O(T^2)$. If $u={\theta }^{\prime }(0)$ is a unit, recursive comparison of coefficients constructs a unique compositional inverse $\psi (T)\in TR[[T]]$ with $\psi (\theta (T))=T=\theta (\psi (T))$. Apply $\psi$ to the homomorphism identity and substitute $X=\psi (U)$, $Y=\psi (V)$ to obtainThus $\psi$ is a homomorphism from $\mathcal{G}$ to $\mathcal{F}$, so $\theta$ is an isomorphism.

Solved by gpt-5.6-sol high.