Solving Polynimals in the ring mod p^r

[JasonRox]

Ok, I'm given this polynomial and I'm asked to find the roots of it in the ring mod p.

And then it asks to do it in, mod p^2, mod p^3 and mod p^4.

I don't remember ever learning how to do it in those powers.

Any tips on how to solve such things?

Note: Without having to sub in all the values. If it was 7^4, that would suck.

Note: Not homework. It's just a problem set to a pdf file I'm going through. It's review for the material, so I want to get a handful of problems down before moving forward.

 

[morphism]

Well if f(x)=0(mod p^k) then f(x)=0(mod p). Maybe this will help.

 

[JasonRox]

Well if f(x)=0(mod p^k) then f(x)=0(mod p). Maybe this will help.

Well, if you look at something mod 49, we can have f(43) = 0 mod 49, but I don't really care that f(43) = 0 mod p. I know that already.

I would still have to check all the numbers up to 49 to see if there are any roots.

 

[Hurkyl]

Well, if you look at something mod 49, we can have f(43) = 0 mod 49, but I don't really care that f(43) = 0 mod p. I know that already.

I would still have to check all the numbers up to 49 to see if there are any roots.

No you don't. You only have to check a few of them...

 

[Jang Jin Hong]

example
if f(k)=0 (mod7)
f(7n+k)=0(mod49)

if you know roots of mod p
and that is denoted as k_1, k_2, k_3, ...
roots of mod p^2 will be
p+k_1, 2p+k_1, 3p+k_1, ... , np+k_1,
p+k_2, 2p+k_2, 3p+k_2, ... , np+k_2,
p+k_3, 2p+k_3, 3p+k_3, ... , np+k_3,
...