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Solving Polynimals in the ring mod p^r

  1. Oct 12, 2008 #1

    JasonRox

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    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.
     
  2. jcsd
  3. Oct 12, 2008 #2

    morphism

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    Well if f(x)=0(mod p^k) then f(x)=0(mod p). Maybe this will help.
     
  4. Oct 13, 2008 #3

    JasonRox

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    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.
     
  5. Oct 13, 2008 #4

    Hurkyl

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    No you don't. You only have to check a few of them....
     
  6. Oct 15, 2008 #5
    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,
    ....
     
    Last edited: Oct 15, 2008
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