Let $p$ be a prime number. Combine Hensel’s lemma and quadratic reciprocity to give a sufficient condition for the general monic quadratic polynomial $$f(X) = X^2 +bX +c ∈\mathbb Z[X]$$ with integer coefficients to have solutions in $\mathbb{Z}/p^n\mathbb{Z}$ for every positive integer $n$.
Hensel's Lemma tells us that for a polynomial $f ∈ \mathbb{Z}[X]$ with integer coefficients, $n ∈ \mathbb{Z}$ with $0 < n$ and $x ∈ Z$ such that
$f(x) = 0$ mod $p^n$ and $f'(x)\neq 0$ mod $p$, we get $y$ such that
$y = x$ mod $p^n$, $f(y) = 0$ mod $p^{n+1}$, and $f'(y)= 0$ mod $p$.
Does that mean that the sufficient condition is simply that $f(x) = 0$ mod $p^n$ and $f'(x)\neq 0$ mod $p$ ?
I am confused. How does that indicate that there is a solution?