If chatacteristic of $K$ is positive, show that every homomorphism from additive group to multiplicative group maps all elements of $K$ to $1$ Suppose $K$ is a field. Denote $(K,+)$ as the abelian group under addition operation and $(K,\times)$ as the abelian group under multiplication opearation. If the characteristic of $K$ is positive, show that every homomorphism $f:(K,+) \rightarrow (K,\times)$ maps all elements of $K$ to $1$.
I don't even know how to start this question. Any hint?

$f(p\cdot a)=f(0)=1$.  And $f(p\cdot a)=f(a)^p$.  So $f(a)^p=1$.  So $0=f(a)^p-1=(f(a)-1)^p$.  So $(f(a)-1)^p=0$. $K$ has no zero divisors so $f(a)-1=0$. So $f(a)=1$.
