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

Repeated formal addition givesfor positive $n$, and the same formulas extend to all $n\in \mathbb{Z}$ using the formal inverse.

Now let $R$ be a field of characteristic $p>0$ and let $h:{\hat{\mathbb{G}}}_a\to {\hat{\mathbb{G}}}_m$ be a homomorphism. Compatibility with multiplication by $p$ givesThe left side is $h(0)=0$, while the Frobenius identity gives the right side asThe formal power series ring over a field is an integral domain, so $h(X{)}^p=0$ implies $h=0$. Hence there are no nonzero homomorphisms from the formal additive group to the formal multiplicative group.