Corollary 1   Let $p$ be a prime number. The ring of $p$ -adic integers $\Ints_p$ contains exactly $p-1$ distinct $(p-1)$ th roots of unity. Furthermore, every $(p-1)$ th root of unity is distinct modulo $p$ .

Proof. Notice that $\Rats_p$ , the $p$ -adic rationals, is a field. Therefore $f(x)=x^{p-1}-1$ has at most $p-1$ roots in $\Rats_p$ (see this entry). Moreover, if we let $a\in \Ints$ with $1\leq a \leq p-1$ then $f(a)=a^{p-1}-1\equiv 0 \mod p$ by Fermat's little theorem. Since $f'(a)=(p-1)\cdot a^{p-2}$ is non-zero modulo $p$ , the trivial case of Hensel's lemma implies that there exist a root of $x^{p-1}-1$ in $\Ints_p$ which is congruent to $a$ modulo $p$ . Hence, there are at least $p-1$ roots in $\Ints_p$ , and we can conclude that there are exactly $p-1$ roots.

Definition 1   The Teichmüller character is a homomorphism of multiplicative groups: $$\omega \colon \mathbb{F}_p^\times \to \Ints_p^\times$$ such that $\omega(a)$ is the unique $(p-1)$ th root of unity in $\Ints_p$ which is congruent to $a$ modulo $p$ (which exists by the corollary above). The map $\omega$ is sometimes called the Teichmüller lift of $\mathbb{F}_p$ to $\Ints_p$ ($0\mod p$ would lift to $0\in \Ints_p$ ).

Remark 1   Some authors define the Teichmüller character to be the homomorphism: $$\hat{\omega}\colon \Ints_p^\times \to \Ints_p^\times$$ defined by $$\hat{\omega}(z)=\lim_{n\to \infty} z^{p^n}.$$ Notice that for any $z\in \Ints_p^\times$ , $\hat{\omega}(z)$ is a $(p-1)$ th root of unity: $$(\hat{\omega}(z))^p=\left( \lim_{n\to \infty} z^{p^n} \right)^p= \lim_{n\to \infty} z^{p^{n+1}}=\hat{\omega}(z).$$ Thus, the value $\hat{\omega}(z)$ is the same than $\omega(z \mod p)$ .