I struggle with this Exercise, or at least the part where one should prove how many solutions there are. Simply inserting f=0 contradicts the suggested number of solutions.
Let $p$ be an odd prime, and let $e\in\mathbb{Z}$ with $e\gt 1$. Let $a$ be an integer of the form $a = p^f b$, where $0\le f \lt e$ and $p \not \mid b$. Consider the integer solutions $z$ to the congruence $z^2 \equiv a \mod{p^e}$. Show that a solution exists if and only if $f$ is even and $b$ is a quadratic residue modulo $p$, in which case there are exactly $2p^f$ distinct solutions modulo $p^e$.
At first I tried to subtract the right hand side and considered the conjugate. Then I factored out $p^{f/2}$, but in the end it didn't work out.
Problem source:
A Computational Introduction to Number Theory and Algebra (Second edition), by Victor Shoup.
Exercise number 2.41
I missunderstood the question, got it know.