Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 136 / 2 / c / Solution

The smallest integer isIndeed, reduction modulo $p$ cannot work: the map $x\mapsto x^p$ is the identity on ${\mathbb{F}}_p^{\times }$, although not every element of ${\mathbb{Z}}_p^{\times }$ is a $p$th power.

For odd $p$, the p-adic unit group decomposes asRaising to the $p$th power is an automorphism on ${\mu }_{p-1}$. On the principal units, the p-adic logarithm identifies it with multiplication by $p$ on $p{\mathbb{Z}}_p$, soHence whether a unit is a $p$th power is determined exactly by its residue modulo $p^2$. Equivalently,This is the pth-power criterion for p-adic units.