How to use Hensels Lemma

Question

Hello I a quite confused in regard to how to use Hansels lemma, especially in some special cases,

for example $$x^2+x+7=0\mod27$$

is equivalent to $$ x^2+x+7=0 \mod 3^3$$

$$x^2+x+7=0 \mod 3$$ has a solution, $x_0=1$

and $$f'(x)=2x+1,$$ so $$f'(1)=3=0\mod 3$$

But the case I had learnt about always had that $f'(x_0) \neq 0 \bmod p$

So in the case that we do have this, what is the general approach?

Answer 1

Let us take your example: $\;f(x)=x^2+x+7=0\pmod{27}\;$ . Doing all the time arithmetic modulo $\;27\;$ , this quadratic's discriminant is

$$\Delta=1-4\cdot7=-27=0\implies x^2+x+7=\left(x+\frac12\right)^2=(x+14)^2$$

and you have a unique double root $\;x=-14=13\;$.

We can see in this case that

$$f(1)=0\pmod 9\;,\;\;f'(1)=0\pmod3$$

which fits case (ii) in this paper