Hensel's lemma and the Teichmuller lift

The following standard lemma is proved by Newton iteration.

Lemma 1.12 (Hensel)   If $f \in \mathbb{Z}_p[x]$ is a polynomial and $\beta \in \mathbb{Z}/p\mathbb{Z}$ is a multiplicity one root of $\overline{f}$, then there is a unique lift of $\beta$ to a root of $f$.

For example, consider the polynomial $f(x) = x^{p-1} - 1$. By Fermat's little theorem, it has $p-1$ distinct roots in $\mathbb{Z}/p\mathbb{Z}$, so by Lemma 1.12 there are $p-1$ roots of $f(x)$ in $\mathbb{Z}_p$, i.e., all the $p-1$st roots of unity are elements of $\mathbb{Z}_p$. The Teichmuller lift is the map that sends any $\beta \in (\mathbb{Z}/p\mathbb{Z})^*$ to the unique $(p-1)$st root of unity in $\mathbb{Z}_p^*$ that reduces to it.

The Teichmuller character is the homomorphism

$$\tau: \mathbb{Z}_p^* \to \mathbb{Z}_p^*$$

obtained by first reducing modulo $p$, then sending an element to its Teichmuller lift. The $1$-unit projection character is the homomorphism

$$\langle \bullet \rangle: \mathbb{Z}_p^* \to 1 + p\mathbb{Z}_p$$

given by

$$\langle x \rangle = \frac{x}{\tau(x)}.$$