Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 4 / 8E / iii

The finite prime-power Teichmuller representative has an explicit description. Let $y$ be the unique least nonnegative residue satisfyingRepeated Fermat's little theorem gives $y\equiv x(\mathrm{mod}p)$. Part (ii) givesso this residue has both required properties.

For uniqueness, suppose $z$ also has them. From $z^p\equiv z(\mathrm{mod}{p}^n)$, raising congruences to the $p$th power repeatedly gives $z^{p^j}\equiv z(\mathrm{mod}{p}^n)$ for every $j\ge 0$. Meanwhile $z\equiv x(\mathrm{mod}p)$ and the lifting result with $r=n-1$ implyHence $z\equiv y(\mathrm{mod}{p}^n)$. Both lie between zero and $p^n-1$, so $z=y$. Thus existence and uniqueness hold for every residue class, including zero, without needing a general Hensel lemma.