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Finding the order of a quotient field

  1. May 2, 2017 #1
    1. The problem statement, all variables and given/known data
    Find the order of ##\mathbb{Z}_3 [x] / \langle x^2 + 2x + 2 \rangle ## and ##\mathbb{Z}_3 [x] / \langle x^2 + x + 2 \rangle ##

    2. Relevant equations


    3. The attempt at a solution
    Is there an efficient method for doing this? Is the answer 27 for both? It would seem that both of these consist of elements of the forms ##ax^2 + bx + c + \langle x^2 + 2x + 2 \rangle## or ##ax^2 + bx + c + \langle x^2 + x + 2 \rangle##, and there are three choices for the coefficients a, b, c, so ##3^3 = 27##
     
    Last edited: May 2, 2017
  2. jcsd
  3. May 2, 2017 #2

    andrewkirk

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    Are you sure? How did you derive this?
     
  4. May 2, 2017 #3
    Well I am not sure. But my intuition tells me that all higher order polynomials can be written as lower powers by using the ##x^2 = -2x -2## or ##x^2 = -x -2##
     
  5. May 2, 2017 #4

    andrewkirk

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    What is the maximum possible order of the remainder polynomial one gets from dividing a polynomial by ##x^22+2x+2##?
     
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