Let $\mathbb{Z}_p$ the ring of $p$-adic numbers. It's known that the multiplicative unit group $\mathbb{Z}_p ^\times$ can be set theoretically described as $\bigcup _{1 \le a \le p-1} a+ p\mathbb{Z}_p$.
I want to know how to see that we have also the group isomorphism $\mathbb{Z}_p ^\times \cong C_{p-1} \times (1+ p \mathbb{Z}_p)$.
This is probably more suitable to MathStackExchange, although there may well be researchers in allied areas that aren't aware of this fact. So in that spirit, here's the standard proof. There is an exact sequence $$ 1 \to 1+p\mathbb Z_p \to \mathbb Z_p^\times \to \mathbb F_p^\times \to 1 .$$ So one needs split this exact sequence, since it is well known that $\mathbb F_p^\times$ is cyclic. The idea is that Hensel's lemma tells us that every root $\bar a$ of $X^p-X$ in $\mathbb F_p$ lifts to a unique root $a\in\mathbb Z_p$ satisfying $a\equiv\bar a\pmod p$. Discarding $0$, we see that every element of $\mathbb F_p^\times$ (all of which are roots of $X^{p-1}-1$ by Fermat's little theorem, or more intrinsically, by Lagrange's theorem) lifts to a unique root of $X^{p-1}-1$ in $\mathbb Z_p$. These lifts are $(p-1)$st roots of unity in $\mathbb Z_p^\times$, and this lifting map $\mathbb F_p^\times\to\mu_{p-1}\subset\mathbb Z_p^\times$ is clearly a homomorphism due to the uniqueness of the lift. The lifting map $\mathbb F_p^\times\to\mu_{p-1}\subset\mathbb Z_p^\times$ is called the Teichmuller character, as has been noted by others in the comments.
