Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 5 / a / Solution

A one-dimensional commutative formal group law over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfyingThe identity axiom gives $F(X,Y)=X+Y+$ terms of total degree at least two. SeekAfter $a_2,\dots ,a_{n-1}$ have been chosen, the coefficient of $X^n$ in $F(X,i(X))$ is $a_n$ plus a known expression in the earlier coefficients. There is therefore a unique choice of $a_n$ making it zero. Recursion constructs the formal inverse $i(X)$ with $F(X,i(X))=0$ and $i(X)\equiv -X(\mathrm{mod}{X}^2)$.