
Euler's criterion
"Let p = a prime. Show $\displaystyle x^2$ ≡ a (mod$\displaystyle p^2 $) has 0 solutions if $\displaystyle x^2$ ≡ a (mod p) has 0 solutions, or 2 solutions if $\displaystyle x^2$ ≡ a (mod p) has 2."
I thought I should use Euler's criterion for this(?), which says $\displaystyle x^2$ ≡ $\displaystyle a (mod p^2)$ if and only if $\displaystyle a^{(p1)/2}$ ≡ 1 (mod p), but I don't know where to go from there.
Also, I know that a (mod p) can have only either exactly two solutions or zero solutions (by Lagrange), so the question remains how to map a (mod p) to a (mod $\displaystyle p^2 $)
Thanks !