Finite prime-power Teichmuller representative ($y\equiv x^{p^{n-1}}(\mathrm{mod}{p}^n)$)

For a prime $p$ and $n\ge 1$, the unique residue $y$ modulo $p^n$ satisfying $y\equiv x(\mathrm{mod}p)$ and $y^p\equiv y(\mathrm{mod}{p}^n)$ is the displayed residue. Existence follows from Fermat's little theorem and prime-power contraction under pth powers. Uniqueness follows by raising any other such residue repeatedly to the $p$th power and applying the same contraction to its congruence with $x$. This is the finite-modulus version of a Teichmuller representative, including the zero class.