Units in $\mathbb{Z}_p$

Question

Prove that $a=\sum_{i=0}^{\infty}a_ip^i\in\mathbb{Z}_p^*\iff a_0\neq 0$

$\textit{Hint}$: use Hensel's lemma

For $b=\sum_{i=0}^{\infty}b_ip^i\in\mathbb{Z}_p$, we have that $a$ is a unit $\iff ab=1$ for some $b\in\mathbb{Z}_p\iff$

$$(\sum_{i=0}^n a_ip^i)(\sum_{i=0}^n b_ip^i)\equiv 1(\text{mod }p^{n+1})\,\,\forall n\geq 0$$

By checking these conditions, I've verified that such $b_0, b_1, b_2,...$ can be found if and only if $a_0\neq 0$, which proves the claim.

However, I have no idea how Hensel's lemma can be applied here. What am I missing?

Answer 1

It feels unnecessary to do this using Hensel's lemma, but one argument would go as follows: one implication follows simply because a unit in $ \mathbf Z_p $ must also be a unit in $ \mathbf Z/p \mathbf Z $ by reduction modulo $ p $. The other implication follows, since if $ a \in \mathbf Z_p $ is nonzero modulo $ p $, the polynomial $ aX - 1 \in \mathbf Z_p[X] $ satisfies the conditions of Hensel's lemma, thus its root modulo $ p $ lifts to a unique root in $ \mathbf Z_p $.