Hello I am a little bit confused about something. I have a question:

Find the solutions of $\displaystyle x^2 \equiv 8 \mod{17}$

and use Hensel's Lemma to solve $\displaystyle x^2 \equiv 8 \mod{17^2}$.

But I'm a bit confused, i can use the roots of the first equation and use hensel's lemma to get ALL the roots? I used it and got 2 (90, and 199 solve the 2nd equation) , but $\displaystyle \mathbb{Z}_{ 17^2 }$ is not a field, so might there not be extra roots of $\displaystyle X^2-8$ due to the lack of unique factorisation?

Thanks for any help