= Finite prime-power Teichmuller representative
{title2=$y\equiv x^{p^{n-1}}\pmod{p^n}$}
For a prime $p$ and $n\geq1$, the unique residue $y$ modulo $p^n$ satisfying $y\equiv x\pmod p$ and $y^p\equiv y\pmod{p^n}$ 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.
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