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Irreducible Polynomials

  1. Sep 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Write [itex]P(x) = x^3+2x+3 [/itex] as the product of Irreducible Polynomials over [itex]Z_5[/itex]

    2. Relevant equations

    Polynomial division

    3. The attempt at a solution

    I start out by taking out a factor of [itex]x+3[/itex]

    That is

    [itex]x+3 \div x^3+2x+3 [/itex]


    I get [itex]P(x) = x^2-3x+1 [/itex] which has zero remainder mod 5.


    Is the product of irreducible polynomial [itex](x+3) (x^2-3x+1) [/itex]


    or do I reduce [itex]P(x) = x^2-3x+1 [/itex] by taking out a factor of x+1 ie


    [itex]x+1 \div x^2-3x+1 [/itex]


    I know the irreducible polynomials coefficients should add up to the original degree ,So I have one with degree 1 and the second with degree 2.

    am I on the right track?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 9, 2010 #2

    Office_Shredder

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    Well, a product of irreducible polynomials requires all your polynomials to be irreducible.

    You're on the right track, the question now is: is x+3 irreducible, and is x2-3x+1 irreducible?
     
  4. Sep 9, 2010 #3
    I evaluated

    [itex]x+1 \div x^2-3x+1 [/itex]

    which is [itex]P(X)= x-4[/itex] zero remainder mod 5

    So I have three irreducible Polynomials whose degrees add to three ie


    [itex](x+3)(x+1)(x-4) [/itex]

    Hows that look
     
  5. Sep 9, 2010 #4
    Is there a way to check that my answer is right?
     
  6. Sep 9, 2010 #5

    Dick

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    Multiply your product out and reduce the coefficients mod 5.
     
  7. Sep 9, 2010 #6
    I multiply it out and get

    [itex]x^3-13x-12[/itex] which is [itex]x^3-3x-2[/itex]mod 5 so I'm doing something wrong.
     
  8. Sep 9, 2010 #7

    Dick

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    Don't forget 2=(-3) and 3=(-2) mod 5.
     
  9. Sep 9, 2010 #8
    I think I need some more practice, It can get confusing just doing ordinary polynomial division without having modulo as well
    [itex]x^3+2x+3[/itex]

    thanks for your help
     
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