Definition: $x\in\mathbb{Z}_{n}^{*}$ is a cubic residue if there exists $y\in\mathbb{Z}_{n}^{*}$ s.t. $y^3\equiv x \pmod{n}$.
I have been asked to prove (and I already did) that if $n=pq$, where $p\neq q$ are two primes that satisfy $p\equiv q\equiv 1\pmod{3}$, and $x\in\mathbb{Z}_{n}^{*}$ is a cubic residue, then the equation $y^3\equiv x \pmod{n}$ has exactly $9$ solutions.
I did so by first showing that if $x\in\mathbb{Z}_{p}^{*}$, then $y^3\equiv x\pmod{p}$ has exactly $3$ solutions (again, $p$ is some prime s.t. $p\equiv1\pmod{3}$), and then using the Chinese Remainder Theorem to get the result.
I am curious about how to prove that if $p=q$ (while all other conditions remain as is), then $y^3\equiv x \pmod{n})$ has exactly $3$ solutions. (This statement might be wrong, but I believe this is not the case.)
The group $\mathbb{Z}_{p^2}^{*}$ is cyclic of order $p(p-1)$.
Suppose you have distinct solutions $y,y'$ than their quotient is an elemnt of order $3$. Conversely, if $y$ is a solution and $t$ an element of order $3$ then $yt$ is also a solution.
A cyclic group whose order is divisible by $3$, contains $2$ elements of order $3$.
This leads to $1 +2 =3$ solutions.