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Math Help - congruence classes and roots of a polynomial.

  1. #1
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    congruence classes and roots of a polynomial.

    I am reading a text book on elementary number theory and I am reading a chapter on congruence classes at the moment and im confused as to what it means when it asks/says that a polynomial  f(x)=x^2+1 has two roots in Z_2 but none in Z_3?

    Thanks very much for any help.
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  2. #2
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    Last edited by kira; March 4th 2011 at 07:52 AM. Reason: wrong thought
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  3. #3
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    Quote Originally Posted by hmmmm View Post
    I am reading a text book on elementary number theory and I am reading a chapter on congruence classes at the moment and im confused as to what it means when it asks/says that a polynomial  f(x)=x^2+1 has two roots in Z_2 but none in Z_3?

    Thanks very much for any help.
    A root is a value of x that make the polynomial equal to zero.

    In \mathbb{Z}_{2} there are only two equivalence classes.

    [0],[1]

    f([0])=[0]^2+[1]=[1] \ne [0] and


    f([1])=[1]^2+[1]=[1] +[1]=[2] =[0] so it is a root and the polynomial can be factored mod 2 as

    (x+[1])(x+[1])=x^2+x+x+1 =x^2+[2]x+1=x^2+1

    Now try the same thing in \mathbb{Z}_3
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  4. #4
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    Thanks very much I understand now i was just getting a bit confused.
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  5. #5
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    For those of us who prefer very strict language, a polynomial does NOT have "roots", equations have roots. Better terminology is that the zeros of a polynomial, p(x), are the roots of the equation p(x)= 0.
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  6. #6
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    Thanks, I probably do need to be a lot more precise with my notation/terminology
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