Solving Congruences and the Relationship between (p^{l-1}) and (p^{l})

[mhill]

if we knew how to solve $$f(x)= 0 mod (p^{l-1})$$ (1)

could we solve then $$f(x)= 0 mod (p^{l})$$ for integer 'l'

the idea is, if it were easy to solve $$f(x)= 0 mod (p)$$ then we could easily find a solution to (1) but i do not know how to do it.

 

[CRGreathouse]

It's not even clear that
$$f(x)\equiv0\pmod{p^l}$$
has solutions, let alone that we can easily find them.

 

[Hurkyl]

http://en.wikipedia.org/wiki/Hensel_Lifting

But don't just immediately click on that link! Think about the problem first. Suppose you knew that f(x) had exactly one root modulo p. (Let's say a is that root) Then isn't there a very narrow range of possibilities for a root of f(x) modulo p^2?