I need to prove that the congruence $f(x)=x^3+3x+9 \equiv 0 (\bmod 5^n)$ has only one solutions for every $n \geq 2$.
I checked with Hensel theorem that for $n=2$ there is one solution indeed. I want to use induction and to use Hensel theorem again, so I assumed that for $n$ there is one solution,r, but for using Hensel Lemma for $n+1$ how do i know that $f'(r)\not\equiv 0 (\bmod 5) $?
Thanks!