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Math Help - elementary # theory

  1. #1
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    elementary # theory

    Determine if f(x) is irreducible over Z/pZ. If not, factor it.
    a) f(x) = x^2 + x + 1 p = 2, 3


    b) f(x) = x^3 + 2 p = 3, 5.

    Could someone tell me wat exactly does Z/pZ mean???????and how to do this problem?
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  2. #2
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    pZ is the prime number given multiplied by the integers, so the multples of the prime number.

    Z/pZ is the ring of integers modulo pZ.

    For example, let p=2, then Z/2Z= Z_2= {0,1}. Essentially, you are looking at the remainders when they are divided by 2.
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  3. #3
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    Quote Originally Posted by terr13 View Post
    pZ is the prime number given multiplied by the integers, so the multples of the prime number.

    Z/pZ is the ring of integers modulo pZ.

    For example, let p=2, then Z/2Z= Z_2= {0,1}. Essentially, you are looking at the remainders when they are divided by 2.
    Hey could you please show me how to do the first question...I am still having troubles doing it. I mean there s nothing alike i could find in the book!

    Thanks so much
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  4. #4
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    Quote Originally Posted by felixmcgrady View Post
    Determine if f(x) is irreducible over Z/pZ. If not, factor it.
    a) f(x) = x^2 + x + 1 p = 2, 3
    To see if you can factor this check which elements in Z/pZ are zeros. If p=2 then there are just two elements in Z/pZ and those are the classes [0] and [1]. Note that [0] is not a zero since [0]^2+[0]+[1] = [1]. And [1] is not a zero since [1]^2 + [1]+[1] = [3] = [1]. Therefore it has no zeros. Since it is a quadradic the condition of not having zeros is equivalent to being irreducible. Therefore f(x) is irreducible over Z/pZ for p=2. Now try doing this problem for p=3.
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