Exercise 1.2 of Washington's book "Introduction to Cyclotomic Fields" says
Suppose $p \equiv 1$ (mod $3$). Using the fact that $\mathbb{Z}_p$ contains the cube roots of unity, show that $x^p+y^p \equiv z^p$ (mod $p^n$), $p \not| xyz$, has solutions for each $n \ge 1$.
For instance, let $p=7$. In $\mathbb{Z}_7$, cube roots of unity are $$1,~ 2 + 4\cdot 7 + 6\cdot 7^2 + 3\cdot 7^3 + \cdots,~ 4 + 2\cdot 7 + 0\cdot 7^2 +3\cdot 7^3 + \cdots.$$ Also let $n=2$. Then $x^7+y^7 \equiv z^7$ (mod $7^2$) has a desired integer solution $(x,y,z)=(1,30,31)$.
But I have no idea how to use the existence of third roots of unity to show the existence of certain integer solution.
Thank you in advance.