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Homework Help: Finding a Minimal Polynomial

  1. Aug 1, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the minimal polynomial of a=y^3 in F=Kron(Z/2Z, x^4+x+1). (Calculate the powers of a^2, a^3, and a^4.)

    2. Relevant equations



    3. The attempt at a solution

    I attempted this trying to follow a similar worked problem in my book:

    a=y^3 & y^4=y+1

    Multiply by y^-3: y=y^-2 + y^-3

    Plug in a

    y=a+1

    y^4+y+1 = 0 ... Multiply by y^-4: 1+y^-3+y^-4 = 0

    Plug in a: a+1+a^-1 = a+1+a^2

    So, a satisfies the irreducible polynomial x^2+x+1. Thus, each of the 16 elements of F can be written as a polynomial of degree at most 2 in a and a^2+a+1=0.
    So, F=Kron(Z/2Z, a, x^2+x+1)

    ...did I do this correctly, or am I even close? I'm not sure of the relevance of calculating the powers of a^2, a^3, and a^4 as hinted in the problem statement.
     
  2. jcsd
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