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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.

Solved by gpt-5.6-sol high.

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.