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Polynomial ring

  1. Apr 26, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f(x) = x2 + 1 in Z3[x]. Find the order of the quotient ring Z3[x]/<f>.


    2. Relevant equations



    3. The attempt at a solution

    Note Z3 is a field. Then Z3[x] is euclidean domain.
    Then for any polynomial g(x) can be written as g(x) = p(x).(x2+1) + r(x) where either r(x) = 0 of deg r(x) < 2.
    That is in Z3[x]/ (x2+1),
    we have g(x) +(x2+1) = (p(x).(x2+1) + r(x) )+(x2+1)
    = r(x) + (x2+1)
    That is every polynomial is equalent to either zero polynomial or a polynomial of degree less than 2.
    So we have the elements in Z3[x]/ (x2+1) are of the form r(x) +(x2+1). with deg r(x) <2.
    So elements are of the form ax+b +(x2+1) in Z3[x]/ (x2+1), where a, b ranges over the elements of Z3.
    So the number of elements in Z3[x]/ (x2+1) is 3x3 = 9.
    Hence the order of Z3[x]/ (x2+1) = 9.
     
  2. jcsd
  3. Apr 27, 2009 #2
    Good job, nice solution.

    In fact, your quotient ring Z3/<X^2+1> is a field of 9 elements. If you are not able to show this yet, you will be soon.
     
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