Theorem 8.3 (All formal groups are isomorphic). Assuming that:

$char R = 0$

Then any formal group

F

over

R

is isomorphic to

{\hat{𝔾}}_{a}

over

R \otimes ℚ

. More precisely

(i)
There is a unique power series
$\log (T) = T + \frac{a_{2}}{2} T^{2} + \frac{a_{3}}{3} T^{3} + \dots$

with $a_{i} \in R$ such that
$\log (F (X, Y)) = \log (X) + \log (Y) . (∗)$
(ii)
There is a unique power series
$\exp (T) = T + \frac{b_{2}}{2!} + \frac{b_{3}}{3!} T^{3} + \dots$

with $b_{i} \in R$ such that
$\exp (\log (T)) = \log (\exp (T)) = T .$