Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 2 / a / Solution

A one-dimensional commutative formal group law over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfyingAn isomorphism from $F$ to $G$ is a series $h(T)=uT+O(T^2)$ with $u\in R^{\ast }$ and

Over a characteristic-zero field $K$, every such formal group is isomorphic to the additive formal group. Differentiate the associativity identity and define the invariant differentialTermwise integration is possible in characteristic zero; the formal logarithmhas leading term $T$. Invariance of ${\omega }_F$ givesand evaluation at $(0,0)$ removes the integration constant. Hence ${\log }_F(F(X,Y))={\log }_F(X)+{\log }_F(Y)$. Its unit linear coefficient gives a compositional inverse, so it is an isomorphism to $X+Y$. Therefore any two one-dimensional commutative formal groups over $K$ are isomorphic.

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