The <formal additive group> and <formal multiplicative group> have laws
$$
\widehat{\mathbb G}_a(X,Y)=X+Y,
\qquad
\widehat{\mathbb G}_m(X,Y)=X+Y+XY.
$$
If $R$ is an <algebra over a field> whose scalar field is $\mathbb Q$, the series
$$
h(X)=\exp(X)-1
$$
has linear coefficient one and satisfies
$$
h(X+Y)=h(X)+h(Y)+h(X)h(Y).
$$
It is therefore the <exponential isomorphism between the additive and multiplicative formal groups>, with inverse $\log(1+X)$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-136/5/b/solution