# Solving Congruences

1. Oct 11, 2008

### 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.

2. Oct 13, 2008

### CRGreathouse

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

3. Oct 13, 2008

### Hurkyl

Staff Emeritus
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?

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