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Math Help - Special polynomials

  1. #1
    Senior Member I-Think's Avatar
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    Special polynomials

    Give an example of a field F and a polynomial f(x)\in{F[x]} which is not the the zero polynomial but f(c)=0 for all c\inF

    I've tried the fields Z (mod p) where p is prime, but now I don't believe these can work , there are n elements in that field, but the highest attainable power is n-2, so that's a dead end, in my understanding.

    Fields Z, Q, and R also don't work, according to my (possibly incorrect) understanding.
    Can anyone help out please?
    Last edited by I-Think; November 14th 2010 at 12:30 PM.
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  2. #2
    Senior Member roninpro's Avatar
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    I think that \mathbb{Z}_p is a good place to start.

    Why not take f(x)=x^2+x over \mathbb{Z}_2?
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  3. #3
    Senior Member I-Think's Avatar
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    Tried it, but in \mathbb{Z}_2, by Fermat's Little Theorem, I believe x^2\equiv{x} (mod 2)
    So that becomes
    f(x)= x+x=2x and 2\equiv0 (mod 2)

    So that is in fact the zero polynomial in \mathbb{Z}_2
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by I-Think View Post
    Tried it, but in \mathbb{Z}_2, by Fermat's Little Theorem, I believe x^2\equiv{x} (mod 2)
    So that becomes
    f(x)= x+x=2x and 2\equiv0 (mod 2)

    So that is in fact the zero polynomial in \mathbb{Z}_2
    But, by your criterion any polynomial such that p(c)=0 for all c is the zero polyommial. In fact, the zero polynomial is the polynomial p(x)=0+0x+0x^2+\cdots+0x^n. Now, p(x)=x^2+x does not have all coefficients of 0, does it?
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