Definition Let $p$ be a prime number. Let $a$ be an integer not divisible by $p$. Let $m > 1$ be an integer. Suppose that there exists an integer $n_0 \geq 1$ with the following property. $x^m \equiv a$ (mod $p^n$) has an integer solution whenever $n \geq n_0$. Then we say $a$ is an $m$-th power residue with respect to $p$.
Can we prove the following theorem without using $p$-adic numbers?
Theorem Let $p$ be a prime number. Let $m > 1$ be an integer. There exists an iteger $n \geq 1$ with the following property. If $a \equiv 1$ (mod $p^n$), then $a$ is an $m$-th power residue with respect to $p$.
Motivation This theorem and its generalization to algebraic number fields can be used in class field theory. I'm interested in proving CFT without $p$-adic numbers(see here).