Let $p$ be a prime, and $x,n$ be integers with $n\geq1$
I am trying to show that the equations $y \equiv x {\mod{p}} $, and $y^p \equiv y {\mod{p^n}}$ have a unique solution for $y$ in $\mathbb{Z}_{p^n}$.
I have found a solution $y \equiv x^{p^{n-1}}$ by estabilishing the identity $x^{p^n} \equiv x^{p^{n-1}} \mod{p^n}$ but I am struggling to prove that this solution is unique. I have tried supposing that if $y$ is a solution and then $y+a$ is a solution for some $1\leq a<p^n$ and then looking for some contradiction and a few other ideas but nothing has yielded anything useful.
Thanks in advance.
