How to prove that $\mathbb{Z}_p^\times$ is isomorphic to $(\mathbb{Z}_p, +) \times \mathbb{F}_p^\times$?

Question

Deduce that $\mathbb{Z}_p^\times$ is isomorphic to $(\mathbb{Z}_p, +) \times \mathbb{F}_p^\times$. I have already deduced that $(1+p\mathbb{Z}_p, \cdot)$ and $(\mathbb{Z}_p,+)$ are isomorph. But I can't find the wanted isomorphism. Can someone help me?

Answer 1

Hint: You should know a map $\mathbb Z_p \twoheadrightarrow \mathbb F_p$. Does it induce a map on the respective units? Then, what is the kernel of that induced map? Then, does it split, i.e. is there a subgroup in the $p$-adic units which is cyclic of order $p-1$ (think about what that would mean for a $p$-adic number: It is a root of ...)?

As an aside, the isomorphism that you claim to already have is not true for $p=2$ (and is a more intricate result than all my hints above: It is a logarithm map).

As another aside, I'm sure this question has been asked here before and the isomorphism(s) are treated in each and every source on $p$-adics.