Group homomorphism $\mathbb{Z}_p \longrightarrow\mathbb{F}_p$

Question

Let $p$ be a prime number, denote by $\mathbb{Z}_p$ the ring of $p-$adic integers.

Can we define a group homomorphism $\mathbb{Z}_p \longrightarrow\mathbb{F}_p$?

Answer 1

There is a homomorphism, called the reduction modulo $p$, given by $$ \sum_{i\ge 0}a_ip^i\mapsto a_0 \mod p. $$ This is a standard definition, which is used among other things to show that the invertible elements in the ring $\mathbb{Z}_p$ consist of $\mathbb{Z}_p^{\times}=\{\sum_{i\ge 0}a_ip^i \mid a_0\neq 0\}$.